Mathematics · Class 11 Science
Ch 11Integral Calculus — Class 11 Mathematics, concept-first.
Calculus grew out of two independently-motivated 17th century investigations — Sir Isaac Newton (1643–1727) approached it through physics: he built an expression for the area under a curve by considering a momentary increase at a point, so that the fundamental theorem of calculus was, in effect, already built into his…
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
Basic Integration Rules
Because integration reverses differentiation, every derivative formula gives a corresponding standard integral. The core standard results (each carries ):
Most relevant Q&A
- Integrate the following with respect to $x$: (i) $x^{11}$ (ii) $\dfrac{1}{x^{7}}$ (iii) $\sqrt[3]{x^{4}}$ (iv) $\left(x^{5}\right)^{1/8}$Free
- Integrate the following with respect to $x$: (i) $\dfrac{1}{\sin^{2}x}$ (ii) $\dfrac{\tan x}{\cos x}$ (iii) $\dfrac{\cos x}{\sin^{2}x}$ (iv)…Free
- Integrate the following with respect to $x$: (i) $12^{3}$ (ii) $\dfrac{x^{24}}{x^{25}}$ (iii) $e^{x}$Preview
- Integrate the following with respect to $x$: (i) $\left(1+x^{2}\right)^{-1}$ (ii) $\left(1-x^{2}\right)^{-1/2}$Preview
- Integrate the following functions with respect to $x$: (i) $(x+5)^{6}$ (ii) $\dfrac{1}{(2-3x)^{4}}$ (iii) $\sqrt{3x+2}$Free
In previous exams
How often this chapter’s concepts have been examined — real appearance data, never estimated.
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Introduction
Calculus grew out of two independently-motivated 17th century investigations — Sir Isaac Newton (1643–1727) approached it through physics: he built an expression for the area under a curve by consider…
Newton-Leibnitz Integral
Inverse operation pairs. We already work comfortably with pairs of operations that undo each other: addition/subtraction , multiplication/division , and raising to a power / taking a root .
Basic Rules of Integration
Because integration is defined as the reverse of differentiation, every standard derivative formula immediately hands us a matching standard antiderivative — simply read the differentiation rule backw…
+−Exercise 11.1i4 questions
- Q1Integrate the following with respect to $x$: (i) $x^{11}$ (ii) $\dfrac{1}{x^{7}}$ (iii) $\sqrt[3]{x^{4}}$ (iv) $\left(x^{5}\right)^{1/8}$Free
- Q2Integrate the following with respect to $x$: (i) $\dfrac{1}{\sin^{2}x}$ (ii) $\dfrac{\tan x}{\cos x}$ (iii) $\dfrac{\cos x}{\sin^{2}x}$ (iv)…Free
- Q3Integrate the following with respect to $x$: (i) $12^{3}$ (ii) $\dfrac{x^{24}}{x^{25}}$ (iii) $e^{x}$Preview
- Q4Integrate the following with respect to $x$: (i) $\left(1+x^{2}\right)^{-1}$ (ii) $\left(1-x^{2}\right)^{-1/2}$Preview
Integrals of the Form $\int f(ax+b)\,dx$
The pattern that motivates this section. Every standard antiderivative on the table in §11.3 was built for a bare variable .
+−Exercise 11.2i5 questions
- Q1Integrate the following functions with respect to $x$: (i) $(x+5)^{6}$ (ii) $\dfrac{1}{(2-3x)^{4}}$ (iii) $\sqrt{3x+2}$Free
- Q2Integrate the following functions with respect to $x$: (i) $\sin 3x$ (ii) $\cos(5-11x)$ (iii) $\text{cosec}^{2}(5x-7)$Free
- Q3Integrate the following functions with respect to $x$: (i) $e^{3x-6}$ (ii) $e^{8-7x}$ (iii) $\dfrac{1}{6-4x}$Preview
- Q4Integrate the following functions with respect to $x$: (i) $\sec^{2}\dfrac{x}{5}$ (ii) $\text{cosec}(5x+3)\cot(5x+3)$ (iii) $30\sec(2-15x)\t…Preview
- Q5Integrate the following functions with respect to $x$: (i) $\dfrac{1}{\sqrt{1-(4x)^{2}}}$ (ii) $\dfrac{1}{\sqrt{1-81x^{2}}}$ (iii) $\dfrac{1…Preview
Properties of Integrals
Integration behaves linearly with respect to constant multiples and sums/differences of functions — this is what lets a complicated-looking integrand be broken into manageable pieces and integrated te…
+−Exercise 11.3i6 questions
- Q1Integrate the following with respect to $x$: $$(x+4)^{5}+\dfrac{5}{(2-5x)^{4}}-\text{cosec}^{2}(3x-1)$$Free
- Q2Integrate the following with respect to $x$: $$4\cos(5-2x)+9e^{3x-6}+\dfrac{24}{6-4x}$$Free
- Q3Integrate the following with respect to $x$: $$\sec^{2}\dfrac{x}{5}+18\cos 2x+10\sec(5x+3)\tan(5x+3)$$Preview
- Q4Integrate the following with respect to $x$: $$\dfrac{8}{\sqrt{1-(4x)^{2}}}+\dfrac{27}{\sqrt{1-9x^{2}}}-\dfrac{15}{1+25x^{2}}$$Preview
- Q5Integrate the following with respect to $x$: $$\dfrac{6}{1+(3x+2)^{2}}-\dfrac{12}{\sqrt{1-(3-4x)^{2}}}$$Preview
- Q6Integrate the following with respect to $x$: $$\dfrac13\cos\left(\dfrac{x}{3}-4\right)+\dfrac{7}{7x+9}+e^{\frac{x}{5}+3}$$Preview
Simple applications
A change of variable, not of method. Up to now has always been the variable of integration. In applications it is often more natural to integrate with respect to whatever variable the problem is phras…
+−Exercise 11.4i5 questions
- Q1If $f'(x) = 4x-5$ and $f(2)=1$, find $f(x)$.Free
- Q2If $f'(x) = 9x^{2}-6x$ and $f(0)=-3$, find $f(x)$.Free
- Q3If $f''(x) = 12x-6$ and $f(1)=30,\ f'(1)=5$, find $f(x)$.Preview
- Q4A ball is thrown vertically upward from the ground with an initial velocity of $39.2$ m/sec. If the only force considered is that attributed…Preview
- Q5A wound is healing in such a way that $t$ days since Sunday the area of the wound has been decreasing at a rate of $-\dfrac{3}{(t+2)^{2}}$ c…Preview
Methods of Integration
Why integration is harder than differentiation. Differentiation has a constructive character — a derivative is, by definition, the limit and there are systematic laws (sum, product, quotient, chain ru…
Decomposition method
Not every integrand matches one of the standard formulas directly. The decomposition method handles this by splitting a single "hard" integrand into a sum or difference of two or more "easy" pieces, e…
Decomposition by Partial Fractions
When the integrand is a proper rational (algebraic) fraction — a ratio of polynomials with and — and it does not simplify by the direct decomposition tricks of §11.7.1, the standard route is to rewrit…
+−Exercise 11.5i20 questions
- Q1$\dfrac{x^3+4x^2-3x+2}{x^2}$Free
- Q2$\left(\sqrt{x}+\dfrac{1}{\sqrt{x}}\right)^2$Free
- Q3$(2x-5)(36+4x)$Free
- Q4$\cot^2 x+\tan^2 x$Preview
- Q5$\dfrac{\cos 2x-\cos 2\alpha}{\cos x-\cos \alpha}$Preview
- Q6$\dfrac{\cos 2x}{\sin^2 x\cos^2 x}$Preview
- Q7$\dfrac{3+4\cos x}{\sin^2 x}$Preview
- Q8$\dfrac{\sin^2 x}{1+\cos x}$Preview
- Q9$\dfrac{\sin 4x}{\sin x}$Preview
- Q10$\cos 3x\cos 2x$Preview
- Q11$\sin^2 5x$Preview
- Q12$\dfrac{1+\cos 4x}{\cot x-\tan x}$Preview
- Q13$e^{x\log a}e^{x}$Preview
- Q14$(3x+4)\sqrt{3x+7}$Preview
- Q15$\dfrac{8^{1+x}+4^{1-x}}{2^{x}}$Preview
- Q16$\dfrac{1}{\sqrt{x+3}-\sqrt{x-4}}$Preview
- Q17$\dfrac{x+1}{(x+2)(x+3)}$Preview
- Q18$\dfrac{1}{(x-1)(x+2)^{2}}$Preview
- Q19$\dfrac{3x-9}{(x-1)(x+2)(x^{2}+1)}$Preview
- Q20$\dfrac{x^{3}}{(x-1)(x-2)}$Preview
Method of Substitution or Change of Variable
The substitution method mirrors, in reverse, the chain rule used for differentiating a function of a function. If is a differentiable function of , then , i.e.
Important Results
Two substitution patterns come up so often that it pays to know them as ready-made formulas rather than re-deriving them every time.
+−Exercise 11.6i16 questions
- Q1$\dfrac{x}{\sqrt{1+x^{2}}}$Free
- Q2$\dfrac{x^{2}}{1+x^{6}}$Free
- Q3$\dfrac{e^{x}-e^{-x}}{e^{x}+e^{-x}}$Free
- Q4$\dfrac{10x^{9}+10^{x}\log_{e}10}{10^{x}+x^{10}}$Preview
- Q5$\dfrac{\sin\sqrt{x}}{\sqrt{x}}$Preview
- Q6$\dfrac{\cot x}{\log(\sin x)}$Preview
- Q7$\dfrac{\csc x}{\log\left(\tan \frac{x}{2}\right)}$Preview
- Q8$\dfrac{\sin 2x}{a^{2}+b^{2}\sin^{2}x}$Preview
- Q9$\dfrac{\sin^{-1}x}{\sqrt{1-x^{2}}}$Preview
- Q10$\dfrac{\sqrt{x}}{1+\sqrt{x}}$Preview
- Q11$\dfrac{1}{x\log x\log(\log x)}$Preview
- Q12$\alpha\beta x^{\alpha-1}e^{-\beta x^{\alpha}}$Preview
- Q13$\tan x\sqrt{\sec x}$Preview
- Q14$x(1-x)^{17}$Preview
- Q15$\sin^{5}x\cos^{3}x$Preview
- Q16$\dfrac{\cos x}{\cos(x-a)}$Preview
Integration by Parts
Integration by parts is the integral counterpart of the product rule for differentiation, and it is the go-to method whenever the integrand is a product of two functions of different types — or a sing…
Bernoulli's Formula for Integration by Parts
When a product integrand needs integration by parts applied two, three, or more times in a row — most often when one factor is for — repeating the ordinary by-parts formula step by step gets long and…
+−Exercise 11.7i3 questions
- Q1Integrate the following with respect to $x$: (i) $9xe^{3x}$ (ii) $x\sin 3x$ (iii) $25xe^{-5x}$ (iv) $x\sec x\tan x$Free
- Q2Integrate the following with respect to $x$: (i) $x\log x$ (ii) $27x^{2}e^{3x}$ (iii) $x^{2}\cos x$ (iv) $x^{3}\sin x$Preview
- Q3Integrate the following with respect to $x$: (i) $\dfrac{x\sin^{-1}x}{\sqrt{1-x^{2}}}$ (ii) $x^{5}e^{x^{2}}$ (iii) $\tan^{-1}\left(\dfrac{8x…Preview
Integrals of the Form ∫e^{ax}sin(bx)dx and ∫e^{ax}cos(bx)dx
8 QA genuinely different situation arises with integrands like or : applying integration by parts does not make the integral simpler or terminate the way it does for — instead, after two applications, th…
+−Exercise 11.8i2 questions
Integration of Rational Algebraic Functions
32 QRational algebraic integrands built from quadratic expressions (or their square roots) do not yield to the substitution and by-parts methods developed earlier in the chapter directly -- they need thei…
+−Exercise 11.10i3 questions
- Q1Find the integrals of the following: (i) $\dfrac1{4-x^2}$ (ii) $\dfrac1{25-4x^2}$ (iii) $\dfrac1{9x^2-4}$Free
- Q2Find the integrals of the following: (i) $\dfrac1{6x-7-x^2}$ (ii) $\dfrac1{(x+1)^2-25}$ (iii) $\dfrac1{\sqrt{x^2+4x+2}}$Preview
- Q3Find the integrals of the following: (i) $\dfrac1{\sqrt{(2+x)^2-1}}$ (ii) $\dfrac1{\sqrt{x^2-4x+5}}$ (iii) $\dfrac1{\sqrt{9+8x-x^2}}$Preview
+−Exercise 11.11i2 questions
+−Exercise 11.12i2 questions
+−Exercise 11.13i25 questions
- Q1If $\int f(x)dx=g(x)+c$, then $\int f(x)g'(x)dx$ is (1) $\int (f(x))^2 dx$ (2) $\int f(x)g(x)dx$ (3) $\int f'(x)g(x)dx$ (4) $\int (g(x))^2 d…Free
- Q2If $\displaystyle\int \dfrac{3^{\frac1x}}{x^2}dx=k\left(3^{\frac1x}\right)+c$, then the value of $k$ is (1) $\log 3$ (2) $-\log 3$ (3) $-\df…Free
- Q3If $\int f'(x)e^{x^2}dx=(x-1)e^{x^2}+c$, then $f(x)$ is (1) $2x^3-\dfrac{x^2}2+x+c$ (2) $\dfrac{x^3}2+3x^2+4x+c$ (3) $x^3+4x^2+6x+c$ (4) $\d…Free
- Q4The gradient (slope) of a curve at any point $(x,y)$ is $\dfrac{x^2-4}{x^2}$. If the curve passes through the point $(2,7)$, then the equati…Preview
- Q5$\displaystyle\int \dfrac{e^x(1+x)}{\cos^2(xe^x)}\,dx$ is (1) $\cot(xe^x)+c$ (2) $\sec(xe^x)+c$ (3) $\tan(xe^x)+c$ (4) $\cos(xe^x)+c$Preview
- Q6$\displaystyle\int \dfrac{\sqrt{\tan x}}{\sin 2x}\,dx$ is (1) $\sqrt{\tan x}+c$ (2) $2\sqrt{\tan x}+c$ (3) $\dfrac12\sqrt{\tan x}+c$ (4) $\d…Preview
- Q7$\displaystyle\int \sin^3 x\,dx$ is (1) $\dfrac{-3}4\cos x-\dfrac{\cos 3x}{12}+c$ (2) $\dfrac34\cos x+\dfrac{\cos 3x}{12}+c$ (3) $\dfrac{-3}…Preview
- Q8$\displaystyle\int \dfrac{e^{6\log x}-e^{5\log x}}{e^{4\log x}-e^{3\log x}}\,dx$ is (1) $x+c$ (2) $\dfrac{x^3}3+c$ (3) $\dfrac3{x^3}+c$ (4)…Preview
- Q9$\displaystyle\int \dfrac{\sec x}{\sqrt{\cos 2x}}\,dx$ is (1) $\tan^{-1}(\sin x)+c$ (2) $2\sin^{-1}(\tan x)+c$ (3) $\tan^{-1}(\cos x)+c$ (4)…Preview
- Q10$\displaystyle\int \tan^{-1}\sqrt{\dfrac{1-\cos 2x}{1+\cos 2x}}\,dx$ is (1) $x^2+c$ (2) $2x^2+c$ (3) $\dfrac{x^2}2+c$ (4) $-\dfrac{x^2}2+c$Preview
- Q11$\displaystyle\int 2^{3x+5}\,dx$ is (1) $\dfrac{3(2^{3x+5})}{\log 2}+c$ (2) $\dfrac{2^{3x+5}}{2\log(3x+5)}+c$ (3) $\dfrac{2^{3x+5}}{2\log 3}…Preview
- Q12$\displaystyle\int \dfrac{\sin^8 x-\cos^8 x}{1-2\sin^2x\cos^2x}\,dx$ is (1) $\dfrac12\sin 2x+c$ (2) $-\dfrac12\sin 2x+c$ (3) $\dfrac12\cos 2…Preview
- Q13$\displaystyle\int \dfrac{e^x\left(x^2\tan^{-1}x+\tan^{-1}x+1\right)}{x^2+1}\,dx$ is (1) $e^x\tan^{-1}(x+1)+c$ (2) $\tan^{-1}(e^x)+c$ (3) $e…Preview
- Q14$\displaystyle\int \dfrac{x^2+\cos^2x}{x^2+1}\csc^2x\,dx$ is (1) $\cot x+\sin^{-1}x+c$ (2) $-\cot x+\tan^{-1}x+c$ (3) $-\tan x+\cot^{-1}x+c$…Preview
- Q15$\displaystyle\int x^2\cos x\,dx$ is (1) $x^2\sin x+2x\cos x-2\sin x+c$ (2) $x^2\sin x-2x\cos x-2\sin x+c$ (3) $-x^2\sin x+2x\cos x+2\sin x+…Preview
- Q16$\displaystyle\int \sqrt{\dfrac{1-x}{1+x}}\,dx$ is (1) $\sqrt{1-x^2}+\sin^{-1}x+c$ (2) $\sin^{-1}x-\sqrt{1-x^2}+c$ (3) $\log\left|x+\sqrt{1-…Preview
- Q17$\displaystyle\int \dfrac{dx}{e^x-1}$ is (1) $\log|e^x|-\log|e^x-1|+c$ (2) $\log|e^x|+\log|e^x-1|+c$ (3) $\log|e^x-1|-\log|e^x|+c$ (4) $\log…Preview
- Q18$\displaystyle\int e^{-4x}\cos x\,dx$ is (1) $\dfrac{e^{-4x}}{17}[4\cos x-\sin x]+c$ (2) $\dfrac{e^{-4x}}{17}[-4\cos x+\sin x]+c$ (3) $\dfra…Preview
- Q19$\displaystyle\int \dfrac{\sec^2x}{\tan^2x-1}\,dx$ (1) $2\log\left|\dfrac{1-\tan x}{1+\tan x}\right|+c$ (2) $\log\left|\dfrac{1+\tan x}{1-\t…Preview
- Q20$\displaystyle\int e^{-7x}\sin 5x\,dx$ is (1) $\dfrac{e^{-7x}}{74}[-7\sin5x-5\cos5x]+c$ (2) $\dfrac{e^{-7x}}{74}[7\sin5x+5\cos5x]+c$ (3) $\d…Preview
- Q21$\displaystyle\int x^2e^{\frac x2}dx$ is (1) $x^2e^{\frac x2}-4xe^{\frac x2}-8e^{\frac x2}+c$ (2) $2x^2e^{\frac x2}-8xe^{\frac x2}-16e^{\fra…Preview
- Q22$\displaystyle\int \dfrac{x+2}{\sqrt{x^2-1}}\,dx$ is (1) $\sqrt{x^2-1}-2\log\left|x+\sqrt{x^2-1}\right|+c$ (2) $\sin^{-1}x-2\log\left|x+\sqr…Preview
- Q23$\displaystyle\int \dfrac{1}{x\sqrt{(\log x)^2-5}}\,dx$ is (1) $\log\left|x+\sqrt{x^2-5}\right|+c$ (2) $\log\left|\log x+\sqrt{\log x-5}\rig…Preview
- Q24$\displaystyle\int \sin\sqrt x\,dx$ is (1) $2\left(-\sqrt x\cos\sqrt x+\sin\sqrt x\right)+c$ (2) $2\left(-\sqrt x\cos\sqrt x-\sin\sqrt x\rig…Preview
- Q25$\displaystyle\int e^{\sqrt x}dx$ is (1) $2\sqrt x\left(1-e^{\sqrt x}\right)+c$ (2) $2\sqrt x\left(e^{\sqrt x}-1\right)+c$ (3) $2e^{\sqrt x}…Preview
Summary
This chapter's toolkit reduces to a short list of building blocks, best remembered in groups.
Sample & Board Papers
Sample papers and previous-year board questions for this subject.
+−Show 30 questionsHide questions30 questions
- Q1$\displaystyle\int \dfrac{e^x}{e^{2x}+1}\,dx$ is: (a) $\log(e^x+1)+c$ (b) $\log(e^{2x}+1)+c$ (c) $\tan^{-1}(e^{2x})+c$ (d) $\tan^{-1}(e^x)+c…Preview
- Q2$\displaystyle\int \left(\dfrac{x-1}{x+1}\right) dx$ is: (a) $x - 2\log(x+1) + c$ (b) $\log(x+1) + c$ (c) $x + 2\log(x-1) + c$ (d) $\log(x-1…Preview
- Q3Evaluate: $\displaystyle\int x e^{-x}\,dx$.Preview
- Q4Evaluate: $\displaystyle\int \dfrac{e^{\sin^{-1}x}}{\sqrt{1-x^2}}\,dx$.Preview
- Q5Evaluate: $\displaystyle\int \sqrt{x^2 - 4x + 6}\,dx$.Preview
- Q6Evaluate: $\displaystyle\int \log_3 x\,dx$.Preview
- Q7(a) Evaluate: $\displaystyle\int_1^3 x^2\,dx$ as limit of sums. **OR** (b) Bag A contains 5 white, 6 black balls and bag B contains 4 white,…Preview
- Q8$\int \dfrac{\sec x}{\sqrt{\cos 2x}}\,dx$ is: (a) $\tan^{-1}(\cos x)+c$ (b) $\sin^{-1}(\tan x)+c$ (c) $\tan^{-1}(\sin x)+c$ (d) $2\sin^{-1}(…Preview
- Q9Evaluate: $\int (x+3)\sqrt{x+2}\,dx$Preview
- Q10$\displaystyle\int \sqrt{\dfrac{1-x}{1+x}}\, dx$ is: (a) $\sqrt{1-x^2} + \sin^{-1}x + c$ (b) $\sin^{-1}x - \sqrt{1-x^2} + c$ (c) $\log\left|…Preview
- Q11$\displaystyle\int e^{-7x}\cos x\, dx$ is: (a) $\dfrac{e^{-7x}}{50}[7\cos x - \sin x] + c$ (b) $\dfrac{e^{-7x}}{50}[-7\cos x + \sin x] + c$…Preview
- Q12Evaluate: $\displaystyle\int \dfrac{e^{-x}}{16 + 9e^{-2x}}\, dx$Preview
- Q13(a) Evaluate: $\displaystyle\int \dfrac{x^2\tan^{-1}(x^3)}{1+x^6}\, dx$ **OR** (b) Evaluate: $\displaystyle\int \dfrac{2x+1}{\sqrt{x^2+4x+9}…Preview
- Q14If $\int f'(x) e^{x^2}\, dx = (x-1)e^{x^2}+c$, then $f(x)$ is: (a) $x^3+4x^2+6x+c$ (b) $2x^3 - \frac{x^2}{2} + x + c$ (c) $\frac{2x^3}{3} -…Preview
- Q15$\int x^2 e^{\frac{x}{2}}\, dx$ is: (a) $2x^2 e^{\frac{x}{2}} - 8xe^{\frac{x}{2}} + 16e^{\frac{x}{2}} + c$ (b) $x^2 e^{\frac{x}{2}} - 4xe^{\…Preview
- Q16Integrate $(x-11)^7$ with respect to $x$.Preview
- Q17If $f'(x)=4x-5$ and $f(2)=1$, find $f(x)$.Preview
- Q18$\displaystyle\int \frac{\sin\sqrt{x}}{\sqrt{x}}\, dx =$ (a) $-2\sin\sqrt{x} + c$ (b) $2\cos\sqrt{x} + c$ (c) $-2\cos\sqrt{x} + c$ (d) $2\si…Preview
- Q19$\displaystyle\int \sin^3 x\, dx$ is: (a) $-\dfrac{3}{4}\cos x + \dfrac{\cos 3x}{12} + c$ (b) $-\dfrac{3}{4}\cos x - \dfrac{\cos 3x}{12} + c…Preview
- Q20Find the integral of $\dfrac{1}{\sqrt{x^2-4x+5}}$.Preview
- Q21Evaluate: $\displaystyle\int \dfrac{6x+5}{\sqrt{1-4x-4x^2}}\, dx$ **OR** If $y = \sin^{-1}\dfrac{1}{2}\left(\sqrt{1+x}+\sqrt{1-x}\right)$ th…Preview
- Q22$\displaystyle\int\dfrac{\sqrt{\tan x}}{\sin 2x}\,dx$ is: (a) $\dfrac{1}{2}\sqrt{\tan x}+C$ (b) $\sqrt{\tan x}+C$ (c) $\dfrac{1}{4}\sqrt{\ta…Preview
- Q23Evaluate: $\displaystyle\int xe^x\,dx$Preview
- Q24(a) Evaluate: $\displaystyle\int\dfrac{3x+5}{x^2+4x+7}\,dx$ **OR** (b) Show that $\cot\left(7\dfrac{1}{2}^\circ\right)=\sqrt2+\sqrt3+\sqrt4+…Preview
- Q25If $\displaystyle\int \dfrac{3^{1/x}}{x^2}\,dx = k\left(3^{1/x}\right) + c$, then the value of $k$ is: (a) $-\dfrac{1}{\log 3}$ (b) $\log 3$…Preview
- Q26$\displaystyle\int 2^{3x+5}\,dx$ is: (a) $\dfrac{2^{3x+5}}{2\log 3} + c$ (b) $\dfrac{3(2^{3x+5})}{\log 2} + c$ (c) $\dfrac{2^{3x+5}}{3\log 2…Preview
- Q27Integrate with respect to $x$ : $\cos 5x \sin 3x$Preview
- Q28If $\int f(x)dx = g(x) + c$, then $\int f(x)\, g'(x)dx$ is: (a) $\int f'(x)\, g(x)dx$ (b) $\int (f(x))^2 dx$ (c) $\int (g(x))^2 dx$ (d) $\in…Preview
- Q29$\int e^{\sqrt{x}}\, dx =$ (a) $2e^{\sqrt{x}}(1-\sqrt{x})+c$ (b) $2\sqrt{x}(1-e^{\sqrt{x}})+c$ (c) $2e^{\sqrt{x}}(\sqrt{x}-1)+c$ (d) $2\sqrt…Preview
- Q30Evaluate: $\displaystyle\int \dfrac{2x+3}{x^2+3x+7}\, dx$Preview