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

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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)e^x2, 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 (xe^x) + c
(b) sec (xe^x) + c
(c) tan (xe^x) + c
(d) cos (xe^x) + 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 |e^x| – log |e^x – 1| + c
(b) log |e^x| + log |e^x – 1| + c
(c) log |e^x – 1| – log |e^x| + c
(d) log |e^x + 1| – log |e^x| + 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 = \(\sqrt{x}\)