Balbharati Maharashtra State Board Std 12 Commerce Statistics Part 1 Digest Pdf Chapter 6 Definite Integration Miscellaneous Exercise 6 Questions and Answers.

## Maharashtra State Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6

(I) Choose the correct alternative:

Question 1.
$$\int_{-9}^{9} \frac{x^{3}}{4-x^{2}} d x$$ = ________
(a) 0
(b) 3
(c) 9
(d) -9
(a) 0

Question 2.
$$\int_{-2}^{3} \frac{d x}{x+5}$$ = _________
(a) -log($$\frac{8}{3}$$)
(b) log($$\frac{8}{3}$$)
(c) log($$\frac{3}{8}$$)
(d) -log($$\frac{3}{8}$$)
(b) log($$\frac{8}{3}$$)

Question 3.
$$\int_{2}^{3} \frac{x}{x^{2}-1} d x$$ = _________
(a) log($$\frac{8}{3}$$)
(b) -log($$\frac{8}{3}$$)
(c) $$\frac{1}{2}$$ log($$\frac{8}{3}$$)
(d) $$\frac{-1}{2}$$ log($$\frac{8}{3}$$)
(c) $$\frac{1}{2}$$ log($$\frac{8}{3}$$)

Question 4.
$$\int_{4}^{9} \frac{d x}{\sqrt{x}}$$ = ___________
(a) 9
(b) 4
(c) 2
(d) 0
(c) 2

Question 5.
If $$\int_{0}^{a} 3 x^{2} d x=8$$, then a = __________
(a) 2
(b) 0
(c) $$\frac{8}{3}$$
(d) a
(a) 2

Question 6.
$$\int_{2}^{3} x^{4}$$ dx = ________
(a) $$\frac{1}{2}$$
(b) $$\frac{5}{2}$$
(c) $$\frac{5}{211}$$
(d) $$\frac{211}{5}$$
(d) $$\frac{211}{5}$$

Question 7.
$$\int_{0}^{2} e^{x}$$ dx = _______
(a) e – 1
(b) 1 – e
(c) 1 – e2
(d) e2 – 1
(d) e2 – 1

Question 8.
$$\int_{a}^{b} f(x) d x$$ = ________
(a) $$\int_{b}^{a} f(x) d x$$
(b) –$$\int_{a}^{b} f(x) d x$$
(c) –$$\int_{b}^{a} f(x) d x$$
(d) $$\int_{0}^{a} f(x) d x$$
(c) –$$\int_{b}^{a} f(x) d x$$

Question 9.
$$\int_{-7}^{7} \frac{x^{3}}{x^{2}+7} d x$$ = _________
(a) 7
(b) 49
(c) 0
(d) $$\frac{7}{2}$$
(c) 0

Question 10.
$$\int_{2}^{7} \frac{\sqrt{x}}{\sqrt{x}+\sqrt{9-x}} d x$$ = _________
(a) $$\frac{7}{2}$$
(b) $$\frac{5}{2}$$
(c) 7
(d) 2
(b) $$\frac{5}{2}$$

(II) Fill in the blanks:

Question 1.
$$\int_{0}^{2} e^{x} d x$$ = ________
e2 – 1

Question 2.
$$\int_{2}^{3} x^{4} d x$$ = __________
$$\frac{211}{5}$$

Question 3.
$$\int_{0}^{1} \frac{d x}{2 x+5}$$ = ____________
$$\frac{1}{2} \log \left(\frac{7}{5}\right)$$

Question 4.
If $$\int_{0}^{a} 3 x^{2} d x$$ = 8, then a = _________
2

Question 5.
$$\int_{4}^{9} \frac{1}{\sqrt{x}} d x$$ = _________
2

Question 6.
$$\int_{2}^{3} \frac{x}{x^{2}-1} d x$$ = _________
$$\frac{1}{2} \log \left(\frac{8}{3}\right)$$

Question 7.
$$\int_{-2}^{3} \frac{d x}{x+5}$$ = _________
$$\log \left(\frac{8}{3}\right)$$

Question 8.
$$\int_{-9}^{9} \frac{x^{3}}{4-x^{2}} d x$$ = _____________
o

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

Question 1.
$$\int_{a}^{b} f(x) d x=\int_{-b}^{-a} f(x) d x$$
True

Question 2.
$$\int_{a}^{b} f(x) d x=\int_{a}^{b} f(t) d t$$
True

Question 3.
$$\int_{0}^{a} f(x) d x=\int_{a}^{0} f(a-x) d x$$
False

Question 4.
$$\int_{a}^{b} f(x) d x=\int_{a}^{b} f(x-a-b) d x$$
False

Question 5.
$$\int_{-5}^{5} \frac{x^{3}}{x^{2}+7} d x=0$$
True

Question 6.
$$\int_{1}^{2} \frac{\sqrt{x}}{\sqrt{3-x}+\sqrt{x}} d x=\frac{1}{2}$$
True

Question 7.
$$\int_{2}^{7} \frac{\sqrt{x}}{\sqrt{x}+\sqrt{9-x}} d x=\frac{9}{2}$$
False

Question 8.
$$\int_{4}^{7} \frac{(11-x)^{2}}{(11-x)^{2}+x^{2}} d x=\frac{3}{2}$$
True

(IV) Solve the following:

Question 1.
$$\int_{2}^{3} \frac{x}{(x+2)(x+3)} d x$$
Solution:

Question 2.
$$\int_{1}^{2} \frac{x+3}{x(x+2)} d x$$
Solution:
Let I = $$\int_{1}^{2} \frac{x+3}{x(x+2)} d x$$
Let $$\frac{x+3}{x(x+2)}=\frac{A}{x}+\frac{B}{x+2}$$
∴ x + 3 = A(x + 2) + Bx
Put x = 0, we get
3 = A(2) + B(0)
∴ A = $$\frac{3}{2}$$
Put x + 2 = 0, i.e. x = -2, we get
-2 + 3 = A(0) + B(-2)
∴ 1 = -2B
∴ B = $$-\frac{1}{2}$$

Question 3.
$$\int_{1}^{3} x^{2} \log x d x$$
Solution:

Question 4.
$$\int_{0}^{1} e^{x^{2}} \cdot x^{3} d x$$
Solution:

Question 5.
$$\int_{1}^{2} e^{2 x}\left(\frac{1}{x}-\frac{1}{2 x^{2}}\right) d x$$
Solution:

Question 6.
$$\int_{4}^{9} \frac{1}{\sqrt{x}} d x$$
Solution:

Question 7.
$$\int_{-2}^{3} \frac{1}{x+5} d x$$
Solution:

Question 8.
$$\int_{2}^{3} \frac{x}{x^{2}-1} d x$$
Solution:

Question 9.
$$\int_{0}^{1} \frac{x^{2}+3 x+2}{\sqrt{x}} d x$$
Solution:

Question 10.
$$\int_{3}^{5} \frac{d x}{\sqrt{x+4}+\sqrt{x-2}}$$
Solution:

Question 11.
$$\int_{2}^{3} \frac{x}{x^{2}+1} d x$$
Solution:

Question 12.
$$\int_{1}^{2} x^{2} d x$$
Solution:

Question 13.
$$\int_{-4}^{-1} \frac{1}{x} d x$$
Solution:

Question 14.
$$\int_{0}^{1} \frac{1}{\sqrt{1+x}+\sqrt{x}} d x$$
Solution:

Question 15.
$$\int_{0}^{4} \frac{1}{\sqrt{x^{2}+2 x+3}} d x$$
Solution:

Question 16.
$$\int_{2}^{4} \frac{x}{x^{2}+1} d x$$
Solution:

Question 17.
$$\int_{0}^{1} \frac{1}{2 x-3} d x$$
Solution:

Question 18.
$$\int_{1}^{2} \frac{5 x^{2}}{x^{2}+4 x+3} d x$$
Solution:

Question 19.
$$\int_{1}^{2} \frac{d x}{x(1+\log x)^{2}}$$
Solution:

Question 20.
$$\int_{0}^{9} \frac{1}{1+\sqrt{x}} d x$$
Solution: