# Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13

## Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.13

Choose the correct or most suitable answer from given four alternatives.
Question 1.
If $$\int f(x) d x$$ = g(x) + c, then $$\int f(x) g^{\prime}(x) d x$$

Solution:
(a)

Question 2.
If , then the value of k is ……………
(a) log 3
(b) -log 3
(c) $$-\frac{1}{\log ^{3}}$$
(d) $$\frac{1}{\log 3}$$
Solution:
(c)

Question 3.
If $$\int f^{\prime}(x) e^{x^{3}} d x$$ = (x – 1)ex2, then f(x) is …………………

Solution:
(d)

Question 4.
The gradient (slope) of a curve at any point (x, y) is $$\frac{x^{2}-4}{x^{2}}$$. If the curve passes through the point(2, 7), then the equation of the curve is ………….
(a) y = x + $$\frac{4}{x}$$ + 3
(b) y = x + $$\frac{4}{x}$$ + 4
(c) y = x2 + 3x + 4
(d) y = x2 – 3x + 6
Solution:

Question 5.

(a) cot (xex) + c
(b) sec (xex) + c
(c) tan (xex) + c
(d) cos (xex) + c
Solution:
(c)

Question 6.
$$\int \frac{\sqrt{\tan x}}{\sin 2 x} d x$$ is ……………..

Solution:
(a)

Question 7.
$$\int \sin ^{3} x d x$$ is …………….

Solution:
(c)
Hint: sin3x = $$\frac{1}{4}$$ (3 sin x – sin 3x)

Question 8.

Solution:
(b)

Question 9.

(a) tan-1 (sin x) + c
(b) 2 sin-1 (tan x) + c
(c) tan-1 (cos x) + c
(d) sin-1 (tan x) + c
Solution:
(d)

= sin-1 (t) + c
= sin-1 (tan x) + c

Question 10.

(a) x2 + c
(b) 2x2 + c
(c) $$\frac{x^{2}}{2}$$ + c
(d) $$-\frac{x^{2}}{2}$$ + c
Solution:
(c)

Question 11.
$$\int 2^{3 x+5} d x$$ is ……………

Solution:
(d)

Question 12.

Solution:
(b)

Question 13.

Solution:
(d)

Question 14.
$$\int \frac{x^{2}+\cos ^{2} x}{x^{2}+1}$$ cosec2xdx is …………….
(a) cot x + sin-1 x + c
(b) -cot x + tan-1 x + c
(c) -tan x + cot-1 x + c
(d) -cot x – tan-1 x + c
Solution:
(d)

Question 15.
$$\int x^{2} \cos x d x$$ is ……………
(a) x2 sin x + 2x cos x – 2 sin x + c
(b) x2 sin x – 2x cos x – 2 sin x + c
(c) -x2 sin x + 2x cos x + 2 sin x + c
(d) -x2 sin x – 2x cos x + 2 sin x + c
Solution:
(a)
Hint:
∫x2 cos x dx
By integration by parts
Let I = ∫x2 cos x dx
u = x2 , dv = cos x dx
du = 2x dx , v =∫ cos x dx = sin x
∫u dv = uv – ∫v du
∫x2 cos x dx = x2 sin x – ∫ sin x 2x dx
= x2 sin x – 2 ∫ x sin x dx
Again using Integration by parts
Take u = x, dv = sin x dx
du = dx, v = ∫sin x dx ⇒ v = – cos x
∫x2 cos dx = x2 sin x – 2[x × – cos x – ∫- cos x dx
= x2 sin x + 2x cos x – 2 ∫cos x dx
= x2 sin x + 2x cos x – 2 sin x + c

Question 16.

Solution:
(b)

Question 17.
$$\int \frac{d x}{e^{x}-1}$$ is …………….
(a) log |ex| – log |ex – 1| + c
(b) log |ex| + log |ex – 1| + c
(c) log |ex – 1| – log |ex| + c
(d) log |ex + 1| – log |ex| + c
Solution:
(c)

Question 18.

Solution:
(b)
We know that

Question 19.

Solution:
(d)

Question 20.

Solution:
(a)
We know that

Question 21.

Solution:
(c)
Hint:

By Bernoulli’s formula,

Question 22.

Solution:
(d)

Question 23.

Solution:
(c)

Question 24.

Solution:
(a)

Question 25.

Solution:
(d)
Hint: Let I = $$\int e^{\sqrt{x}} d x$$
t = $$\sqrt{x}$$