TS Inter 2nd Year Maths 2B Integration Important Questions

Students must practice these TS Inter 2nd Year Maths 2B Important Questions Chapter 6 Integration to help strengthen their preparations for exams.

TS Inter 2nd Year Maths 2B Integration Important Questions

Question 1.
Find ∫2x⁷ dx on R.
Solution:

Question 2.
Evaluate ∫cot²xdx on l⊂R {nπ:n∈Z)
Solution:
∫cot²xdx =∫(cosec²x – 1)dx
= ∫ cosec² x – ∫dx = – dx = – cotx – x + c

Question 3.
Evaluate \(\int\left(\frac{x^6-1}{1+x^2}\right)\) dx for x ∈ R
Solution:

Question 4.
Find ∫(1 – x)(4 -3x) (3 + 2x) dx ; x ∈R.
Solution:
(1- x)(4 – 3x)(3+2x) = 6x³ – 5x² – 13x + 12
∴ ∫ (1 – x)(4 – 3x)(3+2x)dx
=∫(6x³– 5x²_13x+ 12)dx

Question 5.
Evaluate \(\int\left(x+\frac{1}{x}\right)^3 d x, x>0\)
Solution:

Question 6.
Find \(\int \sqrt{1+\sin 2 x}\) dx on R.
Solution:

Question 7.
Find \(\int \frac{6 x}{3 x^2-2}\) dx on any interval I ⊂ R \(\left\{-\sqrt{\frac{2}{3}}, \sqrt{\frac{2}{3}}\right\}\)
Solution:

Question 8.
\(\int \frac{\left(\sin ^{-1} x\right)^2}{\sqrt{1-x^2}}\) dx on R
Solution:

Question 9.
Evaluate \(\int \frac{1}{1+(2 x+1)^2}\) dx on R.
Solution:

Question 10.
Evaluate \(\int \frac{x^5}{1+x^{12}}\) dx on R
Solution:

Question 11.
∫ cos³ sinx dx on R.
Solution:

Question 12.
Find \(\int\left(1-\frac{1}{x^2}\right) e^{\left(x+\frac{1}{x}\right)}\) dx on I where I = (0,∞)
Solution:

Question 13.
Evaluate \(\int \frac{1}{\sqrt{\sin ^{-1} x} \sqrt{1-x^2}}\)
Solution:

Question 14.
Evaluate \(\int \frac{\sin ^4 x}{\cos ^6 x} d x \), x ∈ I ⊂ R – {(2n+1) \(\frac{\pi}{2}\) …………… n∈Z}
Solution:

Question 15.
Evaluate ∫ sin² x dx on R.
Solution:

Question 16.
Find \(\int \frac{x^2}{\sqrt{x+5}}\) on (-5, ∞)
Solution:

Question 17.
Find  \(\int \frac{x}{\sqrt{1-x}}\) dx, x∈1=(0,1)
Solution:
Let 1 – x = t² over (0, 1)
then – dx = 2t dt and x = 1 – t²

Question 18.
Evaluate \(\int \frac{d x}{(x+5) \sqrt{x+4}}\) on (-4,∞)
Solution:
Let x + 4 – t² then dx – 2t dt
defined over (- 4, ∞)

Question 19.
Evaluate \(\int \frac{d x}{\sqrt{4-9 x^2}} \text { on } I=\left(-\frac{2}{3}, \frac{2}{3}\right)\)
Solution:

Question 20.
\(\int \frac{1}{a^2-x^2}\) dx for x E I = (- a, a).
Solution:

Question 21.
Evaluate \(\int \frac{1}{1+4 x^2}\) dx on R.
Solution:

Question 22.
Evaluate \(\int \sqrt{4 x^2+9}\) dx on R.
Solution:

Question 23.
Evaluate \(\int \frac{1}{\sqrt{4-x^2}}\) dx on (-2,2)
Solution:

Question 24.
Evaluate \(\int \sqrt{9 x^2-25} d x \text { on }\left(\frac{5}{3}, \infty\right)\)
Solution:

Question 25.
Evaluate \(\int \sqrt{16-25 x^2} d x \text { on }\left(-\frac{4}{5}, \frac{4}{5}\right)\)
Solution:

Question 26.
Find \(e^x \frac{(1+x)}{(2+x)^2}\) dx on l ⊂ R – {-2}
Solution:

Question 27.
Evaluate \(\int \frac{d x}{\sqrt{x^2+2 x+10}}\)
Solution:

Question 28.
Evaluate \(\int \frac{d x}{\sqrt{1+x-x^2}}\)
Solution:

Short Answer Type Questions

Question 1.
Evaluale \(\int\left(\frac{2 x^3-3 x+5}{2 x^2}\right)\) dx for x>0 and verify the result by differentiation.
Solution:

This is the given expression and the result is correct.

Question 2.
Evaluate \(\int \frac{1}{a \sin x+b \cos x}\) dx where a, b ∈ R and a² + b ² ≠ 0 on R
Solution:

Question 3.
Evaluate ∫ x sin^-1 x dx on (-1, 1).
Solution:
We use integration by parts by suitably
choosing y x and u = sin^-1 x so that

Question 4.
Evaluate ∫ x²cosx dx dx
solution:
Use integration by parts by choosing u x² and y = cos x, we get
∫ x² cos x dx dx = x² ∫cos x dx
\(-\int\left[\frac{d}{d x}\left(x^2\right) \int \cos x d x\right] dx\)
⇒ x² sin x – f 2x sin x dx
⇒ x² sinx-[2x(- cosx) – ∫2(- cosx)dx]
⇒ x² sinx+ 2xcosx – 2sinx+c
⇒ (x² -2) sinx + 2xcosx+ c
(again using integration by parts on ∫ 2x sin x dx)

Question 5.
Evaluate ∫ e^x sinx dx on R.
Solution:
Let I = ∫ e^x sinx dx. Then using integration by parts by taking u = e^x v = sin x we get

Question 6.
Find ∫ e^ax cos(bx +c) dx on R, where a,b,c are real numbers and b ≠ 0.
Solution:
Let I =∫ e^ax cos(bx +c) dx
using integration by parts by suitably choosing e^ax = u and cos (bx + c) = v, we get

Question 7.
Evaluate \(\int \tan ^{-1} \sqrt{\frac{1-x}{1+x}}\) dx on (-1,1)
Solution:

Question 8.
Evaluate \(\int e^x\left(\frac{1-\sin x}{1-\cos x}\right)\) dx on I⊂R {2nπ : n ∈Z}.
Solution:

Question 9.
\(\int \frac{d x}{(x+5) \sqrt{x+4}}\)
Solution:

Question 10
Evaluate \(\int \frac{d x}{5+4 \cos x}\)
Solution:

Question 11.
\(\int \frac{d x}{3 \cos x+4 \sin x+6}\)
Solution:

Question 12.
Find \(\int \frac{d x}{d+e \tan x}\)
Solution:

Question 13.
Evaluate \(\int\left(\frac{\cos x+3 \sin x+7}{\cos x+\sin x+1}\right) d x\)
Solution:

Question 14.
Find \(\int \frac{x^3-2 x+3}{x^2+x-2}\) dx
Solution:
Integrand is a rational function in which the degree of the numerator Is greater than the denominator. Hence using synthetic division.

Question 15.
Find \(\int \frac{2 x^2-5 x+1}{x^2\left(x^2-1\right)}\) dx
Solution:

Question 16.
Find \(\int \frac{3 x-5}{x\left(x^2+2 x+4\right)}\) dx
Solution:

Long Answer Type Questions

Question 1.
Evaluate \(\int \tan ^{-1}\left(\frac{2 x}{1-x^2}\right)\) dx on l ⊂ R {-1,1}
Solution:

Question 2.
Find \(\int \frac{x^2 e^{m \sin ^{-1} x}}{\sqrt{1-x^2}}\) dx on (-1,1) where m is a real number.
Solution:

Question 3.
Evaluate \(\int \frac{x+1}{x^2+3 x+12}\) dx
Solution:

Question 4.
Evaluate \(\int(3 x-2) \sqrt{2 x^2-x+1}\) dx
Solution:

Question 5.
Evaluate \(\int \frac{2 x+5}{\sqrt{x^2-2 x+10}}\) dx
Solution:

Question 6.
Evaluate \(\int \frac{2 x+1}{x\left(x^2+4\right)^2}\) dx
Solution:

Question 7.
Find reduction formula for ∫ xⁿ e^ax dx, n being a positive integer and hence evaluate dx
Solution:

Question 8.
Obtain reduction formula for ∫ sinⁿ x dx for an integer n ≥ 2 and hence obtain ∫ sin⁴ xdx.
Solution:

Question 9.
Obtain reduction formula for ∫ sin^m x cosⁿ x dx for a positive integer m and integer n≥ 2.
Solution:

Question 10.
Obtain reduction formula for ∫ tanⁿ x dx for an integar n ≥ 2 and hence find ∫ tan⁶ x dx.
Solution:

Question 11.
Obtain reduction formula for ∫ secⁿ x dx for n ≥ 2 and hence evaluate ∫ sec⁵ xdx
Solution: