Balbharati Maharashtra State Board Std 12 Commerce Statistics Part 1 Digest Pdf Chapter 5 Integration Miscellaneous Exercise 5 Questions and Answers.

## Maharashtra State Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5

(I) Choose the correct alternative from the following:

Question 1.
The value of $$\int \frac{d x}{\sqrt{1-x}}$$ is
(a) 20$$\sqrt{1-x}$$ + c
(b) -2$$\sqrt{1-x}$$ + c
(c) √x + c
(d) x + c
(b) -2$$\sqrt{1-x}$$ + c

Question 2.
$$\int \sqrt{1+x^{2}} d x$$ =
(a) $$\frac{x}{2} \sqrt{1+x^{2}}+\frac{1}{2} \log \left(x+\sqrt{1+x^{2}}\right)+c$$
(b) $$\frac{2}{3}\left(1+x^{2}\right)^{3 / 2}+c$$
(c) $$\frac{1}{3}\left(1+x^{2}\right)+c$$
(d) $$\frac{(x)}{\sqrt{1+x^{2}}}+c$$
(a) $$\frac{x}{2} \sqrt{1+x^{2}}+\frac{1}{2} \log \left(x+\sqrt{1+x^{2}}\right)+c$$

Question 3.
$$\int x^{2}(3)^{x^{3}} d x$$ =
(a) $$\text { (3) }^{x^{3}}+c$$
(b) $$\frac{(3)^{x^{3}}}{3 \cdot \log 3}+c$$
(c) $$\log 3(3)^{x^{3}}+c$$
(d) $$x^{2}(3)^{x 3}$$
(b) $$\frac{(3)^{x^{3}}}{3 \cdot \log 3}+c$$
Hint:
Put x3 = t

Question 4.
$$\int \frac{x+2}{2 x^{2}+6 x+5} d x=p \int \frac{4 x+6}{2 x^{2}+6 x+5} d x$$ + $$\frac{1}{2} \int \frac{d x}{2 x^{2}+6 x+5}$$, then p = __________
(a) $$\frac{1}{3}$$
(b) $$\frac{1}{2}$$
(c) $$\frac{1}{4}$$
(d) 2
(c) $$\frac{1}{4}$$
Hint:
$$\int \frac{x+2}{2 x^{2}+6 x+5} d x=\int \frac{\frac{1}{4}(4 x+6)+\frac{1}{2}}{2 x^{2}+6 x+5} d x$$

Question 5.
$$\int \frac{d x}{\left(x-x^{2}\right)}$$ = ________
(a) log x – log(1 – x) + c
(b) log(1 – x2) + c
(c) -log x + log(1 – x) + c
(d) log(x – x2) + c
(a) log x – log(1 – x) + c

Question 6.
$$\int \frac{d x}{(x-8)(x+7)}$$ = __________
(a) $$\frac{1}{15} \log \left|\frac{x+2}{x-1}\right|+c$$
(b) $$\frac{1}{15} \log \left|\frac{x+8}{x+7}\right|+c$$
(c) $$\frac{1}{15} \log \left|\frac{x-8}{x+7}\right|+c$$
(d) (x – 8)(x – 7) + c
(c) $$\frac{1}{15} \log \left|\frac{x-8}{x+7}\right|+c$$

Question 7.
$$\int\left(x+\frac{1}{x}\right)^{3} d x$$ = _________
(a) $$\frac{1}{4}\left(x+\frac{1}{x}\right)^{4}+c$$
(b) $$\frac{x^{4}}{4}+\frac{3 x^{2}}{2}+3 \log x-\frac{1}{2 x^{2}}+c$$
(c) $$\frac{x^{4}}{4}+\frac{3 x^{2}}{2}+3 \log x+\frac{1}{x^{2}}+c$$
(d) $$\left(x-x^{-1}\right)^{3}+c$$
(b) $$\frac{x^{4}}{4}+\frac{3 x^{2}}{2}+3 \log x-\frac{1}{2 x^{2}}+c$$
Hint:
$$\left(x+\frac{1}{x}\right)^{3}=x^{3}+3 x+\frac{3}{x}+\frac{1}{x^{3}}$$

Question 8.
$$\int\left(\frac{e^{2 x}+e^{-2 x}}{e^{x}}\right) d x$$
(a) $$e^{x}-\frac{1}{3 e^{3 x}}+c$$
(b) $$e^{x}+\frac{1}{3 e^{3 x}}+c$$
(c) $$e^{-x}+\frac{1}{3 e^{3 x}}+c$$
(d) $$e^{-x}-\frac{1}{3 e^{3 x}}+c$$
(a) $$e^{x}-\frac{1}{3 e^{3 x}}+c$$

Question 9.
∫(1 – x)-2 dx = ___________
(a) (1 + x)-1 + c
(b) (1 – x)-1 + c
(c) (1 – x)-1 – 1 + c
(d) (1 – x)-1 + 1 + c
(b) (1 – x)-1 + c

Question 10.
$$\int \frac{\left(x^{3}+3 x^{2}+3 x+1\right)}{(x+1)^{5}} d x$$ = _______
(a) $$\frac{-1}{x+1}+c$$
(b) $$\left(\frac{-1}{x+1}\right)^{5}+c$$
(c) log(x + 1) + c
(d) log|x + 1|5 + c
(a) $$\frac{-1}{x+1}+c$$
Hint:
x3 + 3x2 + 3x + 1 = (x + 1)3

(II) Fill in the blanks.

Question 1.
$$\int \frac{5\left(x^{6}+1\right)}{x^{2}+1}$$dx = x4 + ___x3 + 5x + c.
$$-\frac{5}{3}$$
Hint:
x6 + 1 = (x2 + 1)(x4 – x2 + 1)

Question 2.
$$\int \frac{x^{2}+x-6}{(x-2)(x-1)} d x$$ = x + ______ + c
4 log|x – 1|
Hint:
x2 + x – 6 = (x + 3)(x – 2)

Question 3.
If f'(x) = $$\frac{1}{x}$$ + x and f(1) = $$\frac{5}{2}$$ then f(x) = log x + $$\frac{x^{2}}{2}$$ + _______
2
Hint:

Question 4.
To find the value of $$\int \frac{(1+\log x) d x}{x}$$ the proper substitution is __________
1 + log x = t

Question 5.
$$\int \frac{1}{x^{3}}\left[\log x^{x}\right]^{2} d x$$ = p(log x)3 + c, then p = _______
$$\frac{1}{3}$$
Hint:
$$\frac{1}{x^{3}}\left(\log x^{x}\right)^{2}=\frac{1}{x^{3}}(x \log x)^{2}=\frac{(\log x)^{2}}{x}$$

(III) State whether each of the following is True or False:

Question 1.
The proper substitution for $$\int x\left(x^{x}\right)^{x}(2 \log x+1) d x \text { is }\left(x^{x}\right)^{x}=t$$
True

Question 2.
If ∫x e2x dx is equal to e2x f(x) + c where c is constant of integration, then f(x) is $$\frac{(2 x-1)}{2}$$.
False

Question 3.
If ∫x f(x) dx = $$\frac{f(x)}{2}$$, then f(x) = $$e^{x^{2}}$$.
True

Question 4.
If $$\int \frac{(x-1) d x}{(x+1)(x-2)}$$ = A log|x + 1| + B log|x – 2|, then A + B = 1.
True

Question 5.
For $$\int \frac{x-1}{(x+1)^{3}} e^{x} d x$$ = ex f(x) + c, f(x) = (x + 1)2.
False

(IV) Solve the following:

1. Evaluate:

(i) $$\int \frac{5 x^{2}-6 x+3}{2 x-3} d x$$
Solution:

(ii) $$\int(5 x+1)^{\frac{4}{9}} d x$$
Solution:

(iii) $$\int \frac{1}{2 x+3} d x$$
Solution:

(iv) $$\int \frac{x-1}{\sqrt{x+4}} d x$$
Solution:

(v) If f'(x) = √x and f(1) = 2, then find the value of f(x).
Solution:
By the definition of integral

(vi) ∫|x| dx if x < 0
Solution:
∫|x| dx = ∫-x dx …..[∵ x < 0]
= -∫x dx
= $$-\frac{x^{2}}{2}$$ + c

2. Evaluate:

(i) Find the primitive of $$\frac{1}{1+e^{x}}$$
Solution:
Let I be the primitive of $$\frac{1}{1+e^{x}}$$

(ii) $$\int \frac{a e^{x}+b e^{-x}}{\left(a e^{x}-b e^{-x}\right)^{2}} d x$$
Solution:

(iii) $$\int \frac{1}{2 x+3 x \log x} d x$$
Solution:

(iv) $$\int \frac{1}{\sqrt{x}+x} d x$$
Solution:

(v) $$\int \frac{2 e^{x}-3}{4 e^{x}+1} d x$$
Solution:

3. Evaluate:

(i) $$\int \frac{d x}{\sqrt{4 x^{2}-5}} d x$$
Solution:

(ii) $$\int \frac{d x}{3-2 x-x^{2}} d x$$
Solution:

(iii) $$\int \frac{d x}{9 x^{2}-25}$$
Solution:

(iv) $$\int \frac{e^{x}}{\sqrt{e^{2 x}+4 e^{x}+13}} d x$$
Solution:

(v) $$\int \frac{d x}{x\left[(\log x)^{2}+4 \log x-1\right]}$$
Solution:

(vi) $$\int \frac{d x}{5-16 x^{2}}$$
Solution:

(vii) $$\int \frac{d x}{25 x-x(\log x)^{2}}$$
Solution:

(viii) $$\int \frac{e^{x}}{4 e^{2 x}-1} d x$$
Solution:

4. Evaluate:

(i) ∫(log x)2 dx
Solution:

(ii) $$\int e^{x} \frac{1+x}{(2+x)^{2}} d x$$
Solution:

(iii) ∫x e2x dx
Solution:

(iv) ∫log(x2 + x) dx
Solution:

(v) $$\int e^{\sqrt{x}} d x$$
Solution:

(vi) $$\int \sqrt{x^{2}+2 x+5} d x$$
Solution:

(vii) $$\int \sqrt{x^{2}-8 x+7} d x$$
Solution:

5. Evaluate:

(i) $$\int \frac{3 x-1}{2 x^{2}-x-1} d x$$
Solution:

(ii) $$\int \frac{2 x^{3}-3 x^{2}-9 x+1}{2 x^{2}-x-10} d x$$
Solution:

(iii) $$\int \frac{1+\log x}{x(3+\log x)(2+3 \log x)} d x$$
Solution: