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Applied Mathematics · Class 12 Commerce

Ch 4Integration and Its Application — Class 12 Applied Mathematics, concept-first.

This concept map places integration within calculus. Differential calculus (from the previous chapter) gives slopes of tangents, rates of change and maxima/minima; integral calculus — the focus of this chapter — covers the methods of integration (by substitution, by partial fractions and by parts), the definite integra…

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Polynomial Integration

Think of integration as the reverse of differentiation. If differentiation tells you the slope of a curve at every point, integration tells you the area under that curve between two points.

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

Concept Map

This concept map places integration within calculus. Differential calculus (from the previous chapter) gives slopes of tangents, rates of change and maxima/minima; integral calculus — the focus of thi…

3.1

Introduction

You already know how to differentiate a function — for instance, the derivative of with respect to is .

+Worked Examplesi2 questions
  1. Example 1Evaluate the following Integrals: (a) $\int (x+3)(x+2)\,dx$ (b) $\int \frac{x^3+1}{x^2}\,dx$ (c) $\int \left[\sqrt{x}+\frac{1}{\sqrt{x}}\rig…Free
  2. Example 2The marginal revenue of a company is given by $MR = 80+20x+3x^2$, where $x$ is the number of units sold for a period. Find the total revenue…Preview
3.2

Integration by Substitution

10 Q

Not every integral can be evaluated directly from the standard formulas — many integrands are composite functions in disguise, and integration by substitution is the technique for simplifying them by…

+Worked Examplesi4 questions
  1. Example 3Evaluate the following: (a) $\int \sqrt{2x+3}\,dx$ (b) $\int e^{4-5x}\,dx$ (c) $\int (ax+b)^2\,dx$ (d) $\int \left[a^x+a^{-x}\right]^2 dx$Free
  2. Example 4Evaluate the following integrals by the method of substitution: (a) $\int x\sqrt{x+2}\,dx$ (b) $\int \frac{x}{\sqrt{x-1}}\,dx$ (c) $\int 3^{…Free
  3. Example 5Evaluate: (a) $\int \frac{1}{\sqrt{9+4x^2}}\,dx$ (b) $\int \frac{1}{\sqrt{5+4x+x^2}}\,dx$Preview
  4. Example 6The weekly marginal cost of producing $x$ pairs of tennis shoes is given by $MC = 17 + \frac{200}{x+1}$, where $C(x)$ is cost in Rupees. If…Preview
+3.16 questions
  1. Q1Evaluate the following: (i) $\int (x^2+1)(x-2)\,dx$ (ii) $\int \left(x+\frac{1}{x}\right)^2 dx$ (iii) $\int \frac{x^3+x^2+x+1}{x+1}\,dx$ (iv…Free
  2. Q2Evaluate the following by substitution method: (i) $\int \frac{x+e^{2x}}{x^2+e^{2x}}\,dx$ (ii) $\int \frac{dx}{\sqrt{x}+x}$ (iii) $\int \fra…Free
  3. Q3Find: (i) $\int \frac{1-2x}{\sqrt{1+x^2}}\,dx$ (ii) $\int \frac{1}{\sqrt{3x^2+2x-1}}\,dx$Preview
  4. Q4If the marginal revenue function of a firm in the production of output is $MR = 40 - 10x^2$ where $x$ is the level of output and total reven…Preview
  5. Q5The marginal cost function of producing $x$ units of a product is given by $MC = \frac{x}{\sqrt{2500+x^2}}$. Find the total cost function an…Preview
  6. Q6The marginal cost of producing $x$ units of a product is given by $MC = x\sqrt{x+1}$. The cost of producing 3 units is ₹7800. Find the cost…Preview
3.3

Integration by Partial Fractions

5 Q

A rational function is a ratio of two polynomials, , with . It is called proper if the degree of is less than the degree of , and improper otherwise.

+Worked Examplesi2 questions
  1. Example 7Identify the following expressions as Rational Functions. Further classify them as Proper or Improper. If Improper, express them as sum of a…Free
  2. Example 8Express the following as sum of two or more partial fractions and hence integrate: (a) $\frac{1}{(x-1)(x+3)}$ (b) $\frac{3x-2}{(x+1)(x-2)^2}…Preview
+3.23 questions
  1. Q1Integrate the following expressions: (i) $\frac{x+1}{(x+2)(x+4)}$ (ii) $\frac{x}{(x^2+1)(x^2+2)}$ (iii) $\frac{1}{e^{2x}-1}$ (iv) $\frac{1}{…Free
  2. Q2The marginal revenue function for a firm is given by $\frac{5x^2+30x+51}{(x+3)^2}$. Show that the revenue function is given by $\frac{2x}{x+…Preview
  3. Q3Find the total revenue function and demand function, if the marginal revenue function is given by $MR(x) = \frac{ab}{(x+b)^2} - c$.Preview
3.4

Integration by Parts

4 Q

When the integrand is a product of two functions, none of the standard formulas or substitution usually applies directly — this is where integration by parts comes in, built by reversing the product r…

+Worked Examplesi2 questions
  1. Example 9Integrate the following: (a) $xe^{2x}$ (b) $\log x$Free
  2. Example 10Evaluate: (a) $\int e^x[x^2+2x]\,dx$ (b) $\int e^x\left[\log x+\frac{1}{x}\right]dx$ (c) $\int e^x\left[\frac{x-1}{x^2}\right]dx$ (d) $\int…Preview
+3.32 questions
  1. Q1Integrate the following functions: (i) $xe^{2x+3}$ (ii) $x\log(x^2+1)$ (iii) $x^2e^x$ (iv) $x\log x$ (v) $x\log 2x$ (vi) $x^2\log x$ (vii) $…Free
  2. Q2Evaluate the following: (i) $\int e^x[x^{-2}-2x^{-3}]\,dx$ (ii) $\int e^{3x}\left[\log 3x+\frac{1}{3x}\right]dx$ (iii) $\int e^{2x}\left[\fr…Preview
3.5

Definite Integral

12 Q

So far we have dealt with indefinite integrals — families of functions differing by a constant . A definite integral, written , is different: it has one fixed numerical value, determined by a lower li…

+Worked Examplesi2 questions
  1. Example 11Evaluate the following definite integrals: (a) $\int_0^3 x^3\,dx$ (b) $\int_0^1 \frac{1}{\sqrt{1+x^2}}\,dx$ (c) $\int_1^4 \frac{x}{(x+1)(x+4…Free
  2. Example 12Evaluate the following: (a) $\int_{-1}^{1} x^2\sqrt{x^3+1}\,dx$ (b) $\int_{-3}^{0} x\sqrt{x+4}\,dx$Preview
+3.410 questions
  1. Q1Evaluate the following definite integral: $\int_e^{e^2} \frac{1}{x\log x}\,dx$Free
  2. Q2Evaluate the following definite integral: $\int_1^2 e^{-\log x}\,dx$Free
  3. Q3Evaluate the following definite integral: $\int_{\log 2}^{\log 4} 2^x\,dx$Free
  4. Q4Evaluate the following definite integral: $\int_0^{\sqrt{3}} \frac{x}{16-x^4}\,dx$Preview
  5. Q5Evaluate the following definite integral: $\int_0^1 \frac{1}{\sqrt{x+1}-\sqrt{x}}\,dx$Preview
  6. Q6Evaluate the following definite integral: $\int_0^1 e^x\sqrt{1+e^x}\,dx$Preview
  7. Q7Evaluate the following definite integral: $\int_4^5 \frac{1}{\sqrt{x^2-16}}\,dx$Preview
  8. Q8Evaluate the following definite integral: $\int_0^1 \log(1+2x)\,dx$Preview
  9. Q9Evaluate the following definite integral: $\int_0^4 \sqrt{x^2+9}\,dx$Preview
  10. Q10Evaluate the following definite integral: $\int_0^1 \frac{3t^2}{(1+t^3)(2+t^3)}\,dt$Preview
3.6

Some Properties of Definite Integrals

6 Q

Definite integrals obey a set of properties that make many integrals far easier to evaluate than working from the fundamental theorem directly, especially when the integrand has some underlying symmet…

+Worked Examplesi4 questions
  1. Example 13Evaluate the following definite integrals: (a) $\int_0^4 |x-2|\,dx$ (b) $\int_{-1}^{2} |x^3-x|\,dx$Free
  2. Example 14Evaluate $\int_{-1}^{1} \frac{e^x}{e^x+e^{-x}}\,dx$Free
  3. Example 15Evaluate $\int_0^1 \frac{\log x}{\log x+\log(1-x)}\,dx$Preview
  4. Example 16Evaluate $\int_1^3 \frac{\sqrt[3]{x}}{\sqrt[3]{x}+\sqrt[3]{4-x}}\,dx$Preview
+3.52 questions
  1. Q1Evaluate the following definite integrals: (i) $\int_0^3 f(x)\,dx$ where $f(x)=\begin{cases} x+1 & x<1 \\ 2x & x\ge 1 \end{cases}$ (ii) $\in…Free
  2. Q2Evaluate $\int_0^2 [x]\,dx$ where $[\,\cdot\,]$ denotes the Greatest integer function.Preview
3.7

Consumers' Surplus and Producers' Surplus

11 Q

In economics, the demand curve plots the relationship between the price of a good and the quantity demanded at that price, over a given period — conventionally, price is measured on the vertical axis…

+Worked Examplesi4 questions
  1. Example 17Find the consumers' surplus for the demand function $p = 25 - x - x^2$ when $p_0 = 19$.Free
  2. Example 18The demand function for a commodity is $p = \frac{10}{x+1}$. Find the consumers' surplus when the prevailing market price is 5.Free
  3. Example 19The supply function for a commodity is $p = x^2 + 4x + 5$ where $x$ denotes supply. Find the producers' surplus when the price is 10.Preview
  4. Example 20Suppose that demand is given by the equation $x_d = 500 - 50P$, where $x_d$ is quantity demanded, and $P$ is the price of the good. Supply i…Preview
+3.67 questions
  1. Q1If the demand function is $p = 35 - 2x - x^2$ and the demand $x_0$ is 3, find the consumers' surplus.Free
  2. Q2If the demand function for a commodity is $p = 25 - x^2$, find the consumers' surplus for $p_0 = 9$.Free
  3. Q3The demand function for a commodity is $p = 10 - 2x$. Find the consumers' surplus for (i) $p = 2$ (ii) $p = 6$.Free
  4. Q4The demand function for a commodity is $p = 80 - 3x - x^2$. Find the consumers' surplus for $p = 40$.Preview
  5. Q5If the supply function is $p = 3x^2 + 10$ and $x_0 = 4$, find the producers' surplus.Preview
  6. Q6If the supply function is $p = 4 - 5x + x^2$, find the producers' surplus when the price is 18.Preview
  7. Q7If the demand and supply curve for computers is $D = 100 - 6P$, $S = 28 + 3P$ respectively where $P$ is the price of computers, what is the…Preview

Summary

This section gathers the key definitions, standard results and formulae of the Integration and Its Application unit of CBSE Class 12 Applied Mathematics (subject code 241) into a single quick-revision…

Case-Based Questions

+Case Based5 questions
  1. Q1The second new species named Puntius euspilurus is an edible freshwater fish found in the Mananthavady river in Wayanad. The epithet euspilu…Free
  2. Q2The second new species named Puntius euspilurus is an edible freshwater fish found in the Mananthavady river in Wayanad. It appears in great…Free
  3. Q3Using the supply schedule and Kerala demand schedule for the fish Puntius euspilurus: | Price P per kg (in ₹) | Quantity (X) of Fish Supplie…Preview
  4. Q4Using the supply schedule and Kerala demand schedule for the fish Puntius euspilurus: | Price P per kg (in ₹) | Quantity (X) of Fish Supplie…Preview
  5. Q5Using the supply schedule and Kerala demand schedule for the fish Puntius euspilurus: | Price P per kg (in ₹) | Quantity (X) of Fish Supplie…Preview

Miscellaneous Exercise

+Miscellaneous4 questions
  1. Q1Integrate the following: (i) $x^3e^{x^2}$ (ii) $\int \frac{(x^4-x)^{1/4}}{x^5}\,dx$ (iii) $\int \frac{x^3+x}{x^4-9}\,dx$ (iv) $\int \frac{2^…Free
  2. Q2Evaluate the following: (i) $\int_2^3 \frac{x^3+1}{x(x-1)}\,dx$ (ii) $\int_{1/3}^{1} \frac{(x-x^3)^{1/3}}{x^4}\,dx$ (iii) $\int_0^1 \log\lef…Free
  3. Q3Show that $\int \frac{(a^x+b^x)^2}{a^x b^x}\,dx = \frac{\left(\frac{a}{b}\right)^x-\left(\frac{b}{a}\right)^x}{\log a-\log b}+2x+C$.Preview
  4. Q4A firm finds that quantity demanded and quantity supplied are 30 units when market price is ₹8 per unit. Further, if price is increased to ₹…Preview