This Integral Calculus – CA Foundation Maths Study Material is designed strictly as per the latest syllabus and exam pattern.

## Integral Calculus – CA Foundation Maths Study Material

**Previous Year Exam Questions**

Question 1.

\(\int_0^1\)(e^{x} + e^{-x}) dx is:

(a) e – e^{-1}

(b) e^{-1} – e

(c) e + e^{1}

(d) None

Solution :

(a) is correct

= e – 1 – e^{-1} + 1 = e – e^{-1}

Question 2.

∫\(\frac{8 x^2}{\left(x^3+2\right)^3}\)dx is equal to: [1 Mark, Nov. 2006]

(a) –\(\frac{4}{3}\)(x^{3} + 2) + C

(b) –\(\frac{4}{3}\)(x^{3} + 2)^{-2} + C

(c) \(\frac{4}{3}\)(x^{3} + 2)^{2} + C

(d) None of these

Answer:

Question 3.

∫\(\frac{d x}{\sqrt{x^2+a^2}}\):

(a) \(\frac{1}{2}\)log(x + \(\sqrt{x^2+a^2}\)) + C

(b) log(x + \(\sqrt{x^2+a^2}\)) + C

(c) log(x\(\sqrt{x^2+a^2}\)) + C

(d) \(\frac{1}{2}\)log(x\(\sqrt{x^2+a^2}\)) + C

Answer:

(b) is correct

I. Remember it as Formula

Trick II. Go by choices

Differentiation of which option is the Integration of given Function

Here to differentiate is more easy than to Integrate.

III Detail Method

Let t – x = \(\sqrt{x^2+a^2}\)

Question 4.

The value of \(\int_0^2 \frac{\sqrt{x}}{\sqrt{x}+\sqrt{2-x}}\)dx is: [1 Mark, Feb. 2007 & May 2007]

(a) 0

(b) 3

(c) 2

(d) 1

Answer:

(d) is correct

2I = 2; I = 1

Question 5.

The Integral of (e^{3x} + e^{-3x})/e^{x} is: [1 Mark, May 2007]

(a) \(\frac{e^{2 x}}{2}+\frac{e^{-4 x}}{4}\) + C

(b) \(\frac{e^{2 x}}{2}-\frac{e^{-4 x}}{4}\) + C

(c) e^{2x} – e^{-4x}

(d) None of these

Answer:

(b) is correct

Where c = Integration (Arbitrary) Constant.

Question 6.

∫x^{2}e^{3x}dx is:

(a) x^{2}.e^{3x} – 2xe^{3x} + 2e^{3x} + C

(b) \(\frac{e^{3 x}}{3}-\frac{x \cdot e^{3 x}}{9}\) + 2e^{3x} + C

(c) \(\frac{x^2 \cdot e^{3 x}}{3}-\frac{2 x \cdot e^{3 x}}{9}+\frac{2}{27}\)e^{3x} + C

(d) None of these

Answer:

(c) is correct.

By parts; we get

Question 7.

\(\int_1^2 \frac{2 x}{1+x^2}\)dx: [1 Mark, May & Aug. 2007]

(a) log_{e}\(\frac{5}{2}\)

(b) log_{e}5 – log_{e}2 + 1

(a) log_{e}\(\frac{2}{5}\)

(d) None of these

Answer:

(a)

= log(1+2^{2}) – log(1 + 1^{2})

= log 5 – log 2 = log\(\frac{5}{2}\)

(a) is correct

Question 8.

The value of \(\int_1^e \frac{(1+\log x)}{x}\)dx is: [Given Loge = 1]. [1 Mark, Aug. 2007]

(a) 1/2

(b) 3/2

(c) 1

(d) 5/2

Answer:

(b) is correct

Question 9.

Find ∫\(\frac{x^3}{\left(x^2+1\right)^3}\)dx: [1 Mark, Aug. 2007]

Answer:

(b) is correct

Question 10.

∫\(\frac{1}{x^2-a^2}\)dx is: [1 Mark, Nov. 2007]

(a) log(x – a) – log(x + a) + C

(b) log x – \(\frac{a}{x+a}\) + C

(c) \(\frac{1}{2 a}\) log\(\left(\frac{x-a}{x+a}\right)\) + C

(d) None of these

Answer:

(c) is correct

Where c = Integration Constant

Question 11.

The value of \(\int_0^1 \frac{d x}{(1+x)(2+x)}\) is : [1 Mark, Nov. 2007]

(a) log \(\frac{3}{4}\)

(b) log \(\frac{4}{3}\)

(c) log 12

(d) None of these

Answer:

(b) is correct

\(\int_0^1 \frac{d x}{(1+x)(2+x)}\)

By Hit & Trial Method; we get

= \(\int_0^1\left(\frac{1}{1+x}-\frac{1}{2+x}\right)\)dx

= [log(1 + x)]^{1}_{0} – [log(2 + x)]^{1}_{0}

= [log(1 + 1)- log(1 + 0)] – [log(2 + 1) – log(2 + 0)]

= log2 – 0 – log3 + log2

= 2log2 – log3

= log2^{2} – log3 = log\(\frac{4}{3}\)

Question 12.

The value of \(\int_2^3\)(5 – x)dx – \(\int_2^3\)f(x)dx is:

Answer:

(b) is correct.

Question 13.

∫\(\frac{e^{\log _{e x} x} d x}{x}\) is:

(a) x^{-1} + C

(b) x + c

(c) x^{2} + C

(d) None

Answer:

(b) is correct

∫\(\frac{e^{\log _e x} d x}{x}\) = ∫\(\frac{x}{x}\)

[Formula a^{x logab} = b^{x}]

= ∫dx = x + c

Question 14.

Evaluate ∫\(\frac{1}{(x-1)(x-2)}\)dx: [1 Mark, June 2008]

(a) log \(\left(\frac{x-2}{x-1}\right)\) + C

(b) log[(x – 2)(x -1)] + C

(c) \(\left(\frac{x-1}{x-2}\right)\) + C

(d) None

Answer:

(a) is correct,

∫\(\frac{1}{(x-1)(x-2)}\) = ∫\(\left(\frac{1}{(x-2)}-\frac{1}{x-1}\right)\)

(By Hit & Trial method)

= log(x – 2)- log(x -1)+ c

= log\(\left(\frac{x-2}{x-1}\right)\) + c

Question 15.

\(\int_1^4\)(2x + 5) dx and the value is: [1 Mark, June 2008]

(a) 10

(b) 3

(c) 30

(d) None

Answer:

(c) Is correct.

= 15 + 15 = 30

Question 16.

∫\(\frac{1}{x\left(x^5-1\right)}\)dx

Answer:

(b) is correct