Mathematics · Class 12 Science
Ch 10Integrals — Class 12 Mathematics, concept-first.
Differentiation takes a function to its derivative ; integration reverses this process -- given a function, it recovers a function whose derivative is the given one. This is why integration is also called antidifferentiation, and an integral in this form (with no upper/lower limits) is called an indefinite integral.
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
Standard Integral Forms
Ten named integral patterns -- , , , and their generalisations to a full quadratic under a denominator or a square root (reduced via completing the square), plus the linear-numerator forms and (numerator split into a mul…
Most relevant Q&A
- Evaluate $\displaystyle\int \frac{2x+3}{\sqrt{x^2+3x+9}}\,dx$.Free
- Evaluate $\displaystyle\int \frac{x+1}{x^2+2x+2}\,dx$.Preview
- Evaluate $\displaystyle\int \frac{dx}{x^2-9}$.Preview
- Evaluate $\displaystyle\int \frac{dx}{x^2+4x+13}$.Preview
- Evaluate $\displaystyle\int \frac{2x+3}{x^2+4x+13}\,dx$.Preview
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.
Integration as the Inverse Process of Differentiation
Differentiation takes a function to its derivative ; integration reverses this process -- given a function, it recovers a function whose derivative is the given one.
Integration by Substitution
When the integrand is not a standard form directly but can be seen as a composite function multiplied by the derivative of its inner function, the method of substitution converts it into a standard in…
Integration by Partial Fractions
A rational function is a ratio of two polynomials. When factors into simpler pieces, the rational function can be rewritten as a SUM of simpler fractions -- its partial fraction decomposition -- each…
Integration by Parts
When the integrand is a PRODUCT of two functions of that is not a simple substitution pattern (i.e. no factor is the derivative of the argument of another), integration by parts is the appropriate tec…
Standard Integral Forms with Algebraic Denominators
Two integrals recur throughout calculus and mechanics: and (the syllabus's ). Both are derived once here and then reused, via completing the square, for every quadratic denominator.
Standard Integral Forms with Square Roots
Paralleling Section 5, the square-root forms and are each derived once by a trigonometric substitution and then reused for every quadratic radicand via completing the square.
Fundamental Theorem of Calculus
So far, has meant the INDEFINITE integral -- a family of antiderivatives. A definite integral is a different object: a specific NUMBER, informally the (signed) area between the curve and the -axis fro…
Properties of Definite Integrals
Beyond direct evaluation via the Fundamental Theorem, definite integrals obey several properties that often shortcut a computation -- particularly useful when an antiderivative is hard to find directl…
Evaluation of Definite Integrals
Evaluating a definite integral combines everything built so far: an indefinite-integration technique (Sections 2-6) to find an antiderivative, the Fundamental Theorem (Section 7) to convert it into a…
Summary
Indefinite integral: where ; ALWAYS verify by differentiating the answer.
Sample & Board Papers
Sample papers and previous-year board questions for this subject.
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- Q1Evaluate: the definite integral from 1 to 2 of x dx / [(x + 1)(x + 2)].Preview
- Q2Evaluate: the indefinite integral of sqrt(1 + sec x) dx. **OR** Evaluate: the indefinite integral of dx / [(x - 1) sqrt(x^2 - 1)].Preview
- Q3Prove that the definite integral from 1 to 3 of dx / [x^2 (x + 1)] = 2/3 + log(2/3).Preview
- Q4Evaluate: lim (n -> infinity) [ n/(n^2+1^2) + n/(n^2+2^2) + ... + 1/(2n) ].Preview
- Q5If f(x) = -f(-x), then the value of the definite integral from -a to a of f(x) dx is equal to (a) 2a (b) a (c) a/2 (d) 0Preview
- Q6Evaluate ∫₁₋₁ x|x| dx (definite integral of x|x| from -1 to 1).Preview
- Q7Evaluate ∫ dx / (x(x² + 1)). **OR** Evaluate ∫ dx / (1 + tan x).Preview
- Q8From the definition of definite integral find the value of ∫₀¹ (2x + 1) dx.Preview
- Q9Evaluate lim(n→∞) [1/(n+1) + 1/(n+2) + ... + 1/(3n)].Preview
- Q10Value of ∫₀^(π/2) cos²x dx is equal to (a) π (b) π/2 (c) π/4 (d) 1Preview
- Q11Show that ∫₀^a f(x)dx = ∫₀^a f(a-x)dx.Preview
- Q12Evaluate ∫√(1+cosec x) dx. **OR** Evaluate ∫(x²+1)eˣ/(x+1)² dx.Preview
- Q13Evaluate ∫₀^(π/4) (sin x + cos x)/(9 + 16 sin 2x) dx.Preview
- Q14Evaluate: lim(n→∞) [1/√(n²-1²) + 1/√(n²-2²) + ... + 1/√(n²-(n-1)²)].Preview
- Q15If ∫₀^x f(t) dt = x + ∫₁^x t f(t) dt, then f(x) is equal to (a) 1+x (b) 1-x (c) 1/(1+x) (d) 1/(1-x)Preview
- Q16Value of ∫₀^π |cos x| dx is equal to (a) 0 (b) 1/2 (c) 1 (d) 2Preview
- Q17If f(x) = -f(-x), show that ∫(-a to a) f(x) dx = 0.Preview
- Q18Evaluate ∫ sin(log x) dx. **OR** Evaluate ∫ (√(cot x) - √(tan x)) dx.Preview
- Q19Show that ∫₀^π (x sin x)/(1 + cos²x) dx = π²/4.Preview
- Q20Evaluate lim(n→∞) [1²/(n³+1³) + 2²/(n³+2³) + ... + 1/2n].Preview
- Q21Evaluate ∫₀² (3x² + 2x) dx as a limit of sum.Preview
- Q22The value of ∫₀¹ d/dx[sin⁻¹(2x/(1+x²))] dx is (a) 0 (b) π (c) π/2 (d) π/4Preview
- Q23If f(x)+f(a-x)=k (constant), then find the value of ∫₀ᵃ f(x) dx.Preview
- Q24Evaluate: ∫ dx/(secx+cosecx). **OR** Evaluate: ∫ (x-1)/[(x+1)√(x³+x²+x)] dx.Preview
- Q25Evaluate: lim(n→∞) [1²/(n³+1³) + 2²/(n³+2³) + 3²/(n³+3³) + ... + 1/2n]. **OR** If f(x)=f(a+x), then prove that ∫[a to a+t] f(x) dx is indepe…Preview
- Q26Evaluate: ∫ dx/√(sin³x sin(x+α)).Preview
- Q27For what values of $a$ and $b$ the following expression is correct? $\int \dfrac{dx}{1 + \sin x} = \tan\left(\dfrac{x}{2} + a\right) + b$Preview
- Q28Integrate: $\int \dfrac{dx}{\tan x + \cot x + \sec x + \operatorname{cosec} x}$Preview
- Q29Evaluate: $\int_a^b \dfrac{f(x)\, dx}{f(x) + f(a + b - x)}$.Preview
More questions
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- Example 1Verify that $\displaystyle\int(3x^2+2x)\,dx = x^3+x^2+C$ by differentiating the answer.Free
- Example 2Evaluate $\displaystyle\int 2x\cos(x^2)\,dx$.Free
- Example 3Evaluate $\displaystyle\int \sin^3x\,dx$.Free
- Example 4Evaluate $\displaystyle\int x\sin x\,dx$.Preview
- Example 5Evaluate $\displaystyle\int x^2\ln x\,dx$.Preview
- Example 6Evaluate $\displaystyle\int \frac{3x+1}{(x-1)(x+2)}\,dx$ using partial fractions.Preview
- Example 7Evaluate $\displaystyle\int \frac{dx}{x^2-9}$.Preview
- Example 8Evaluate $\displaystyle\int \frac{dx}{x^2+4x+13}$.Preview
- Example 9Evaluate $\displaystyle\int \frac{2x+3}{x^2+4x+13}\,dx$.Preview
- Example 10Evaluate $\displaystyle\int \frac{dx}{\sqrt{9-x^2}}$.Preview
- Example 11Evaluate $\displaystyle\int \sqrt{9-x^2}\,dx$.Preview
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- Q25Evaluate $\displaystyle\int \frac{dx}{\sqrt{x^2+16}}$.Free
- Q26Evaluate $\displaystyle\int \frac{dx}{\sqrt{x^2+6x+13}}$.Free
- Q27Evaluate $\displaystyle\int \frac{x+3}{\sqrt{x^2+6x+13}}\,dx$.Preview
- Q28Evaluate $\displaystyle\int \sqrt{x^2-9}\,dx$.Preview
- Q29Evaluate $\displaystyle\int \sqrt{x^2+4x+13}\,dx$.Preview
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- Q30Evaluate $\displaystyle\int_1^3 (2x+1)\,dx$ using the Fundamental Theorem of Calculus.Free
- Q31Evaluate $\displaystyle\int_0^{\pi/2} \frac{\sin x}{\sin x+\cos x}\,dx$ using the property $\int_0^af(x)\,dx=\int_0^af(a-x)\,dx$.Free
- Q32Evaluate $\displaystyle\int_{-1}^{1} x^3\cos x\,dx$ by identifying whether the integrand is odd or even.Preview
- Q33Evaluate $\displaystyle\int_{-\pi/2}^{\pi/2} \cos^2x\,dx$ by identifying whether the integrand is odd or even.Preview