Inter 2nd Year Maths 2B Integration Solutions Ex 6(a)

Practicing the Intermediate 2nd Year Maths 2B Textbook Solutions Inter 2nd Year Maths 2B Integration Solutions Exercise 6(a) will help students to clear their doubts quickly.

Intermediate 2nd Year Maths 2B Integration Solutions Exercise 6(a)

I. Evaluate the following integrals.

Question 1.
∫(x³ – 2x² + 3) dx on R.
Solution:
∫(x³ – 2x² + 3) dx = \(\frac{x^4}{4}-\frac{2}{3}\)x³ + 3x + c

Question 2.
∫2x√x dx on (0, ∞).
Solution:
∫2x√x dx = 2 ∫ x3/2 dx = \(\frac{2x^{5/2}}{5/2}\)
= \(\frac{4}{5}\)x5/2 + c

Question 3.
∫\(\sqrt[3]{2 x^2}\) dx’on (0, ∞).
Solution:
∫\(\sqrt[3]{2 x^2}\) dx = ∫ 21/3. x2/3 dx
= 21/3. \(\frac{x^{5/3}}{5/3}\) + c
= \(\sqrt[3]{2}\).\(\frac{3}{5}\)x5/3 + c

Question 4.
∫\(\frac{x^2+3x-1}{2x}\)dx, x ∈ I ⊂ R\{0}.
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(a) 1

Inter 2nd Year Maths 2B Integration Solutions Ex 6(a)

Question 5.
∫\(\frac{1-\sqrt{x}}{x}\)dx on (0, ∞).
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(a) 2

Question 6.
∫(\(1+\frac{2}{x}-\frac{3}{x^2}\)) dx on I⊂R\{0}
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(a) 3

Question 7.
∫(x + \(\frac{4}{1+x^2}\))dx on R.
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(a) 4

Question 8.
∫(ex \(\frac{1}{x}-\frac{2}{\sqrt{x^2+1}}\))dx on I⊂R\[-1, 1].
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(a) 5

Question 9.
∫(\(\frac{1}{1-x^2}+\frac{1}{1+x^2}\))dx on (-1, 1).
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(a) 6

Question 10.
∫(\(\frac{1}{1-x^2}+\frac{2}{1+x^2}\))dx on (-1, 1).
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(a) 7

Question 11.
∫elog(1+tan²x) dx on I ⊂ R \{\(\frac{(2n+1)\pi}{2}\):n ∈ Z}
Solution:
∫elog(1+tan²x) dx = ∫elog(sec²x) dx
= ∫sec²x dx = tan x + c

Inter 2nd Year Maths 2B Integration Solutions Ex 6(a)

Question 12.
∫\(\frac{\sin^{2}x}{1+\cos2x}\) dx on I ⊂ R \{(2n ± 1)π : n ∈ Z}
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(a) 8

II. Evaluate the following intergrals.

Question 1.
∫(1 – x²)³ dx on (-1, 1).
Solution:
∫(1 – x²)³ dx = ∫(1 – 3x² + 3x4 – x6)dx
= x – x³ + \(\frac{3}{5}\)x5 – \(\frac{x^7}{7}\) + c

Question 2.
∫(\(\frac{3}{\sqrt{x}}-\frac{2}{x}+\frac{1}{3x^2}\)) dx on (0, ∞).
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(a) 9

Question 3.
∫(\(\frac{\sqrt{x}+1}{x}\))² dx on (0, ∞).
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(a) 10

Question 4.
∫(\(\frac{(3x+1)^2}{2x}\)) dx, x ∈ I ⊂ R\ {0}.
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(a) 11

Question 5.
∫(\(\frac{2x-1}{3\sqrt{x}}\))² dx on (0, ∞).
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(a) 12

Inter 2nd Year Maths 2B Integration Solutions Ex 6(a)

Question 6.
∫(\(\frac{1}{\sqrt{x}}+\frac{2}{\sqrt{x^2-1}}-\frac{3}{2x^2}\))² dx on (0, ∞).
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(a) 13

Question 7.
∫(sec² x – cos x + x²) dx, x ∈ I ⊂ R/{\(\frac{n \pi}{2}\) : n is an odd integer}.
Solution:
∫(sec² x – cos x + x²) dx
= ∫sec² x dx – ∫cos x + ∫x² dx
= tan x – sin x + \(\frac{x^3}{3}\) + C

Question 8.
∫(sec x tan x + \(\frac{3}{x}\) – 4) dx, x ∈ I ⊂ R\ ({\(\frac{n \pi}{2}\) : n is an odd integer} ∪ {0}).
Solution:
∫(sec x tan x + \(\frac{3}{x}\) – 4) dx
= sec x tan x dx + 3∫\(\frac{dx}{x}\) – 4 ∫dx
= sec x + 3 log |x| – 4x + c

Question 9.
∫(√x – \(\frac{2}{1-x^2}\)) dx on (0, 1).
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(a) 14

Question 10.
∫(x³ – cos x + \(\frac{4}{\sqrt{x^2+1}}\)) dx
Solution:
∫(x³ – cos x + \(\frac{4}{\sqrt{x^2+1}}\)) dx
= ∫ x³ dx – ∫cos x dx + 4 ∫\(\frac{dx}{\sqrt{x^2+1}}\)
= \(\frac{x^4}{4}\) – sin x + 4 sinh-1 x + C

Question 11.
∫(cosh x + \(\frac{1}{\sqrt{x^2+1}}\))dx, x ∈ R.
Solution:
∫(cosh x + \(\frac{1}{\sqrt{x^2+1}}\))dx
= ∫cosh x dx + ∫\(\frac{dx}{\sqrt{x^2+1}}\)
= sinh x + sinh-1 x + c

Question 12.
∫(sinh x + \(\frac{1}{(x^2-1)^{1/2}}\)) dx, x ∈ I ⊂ (-∞, -1) ∪ (1, ∞).
Solution:
∫(sinh x + \(\frac{1}{(x^2 – 1)^{1/2}}\)) dx
= ∫sinh x dx + ∫\(\frac{dx}{\sqrt{x^2-1}}\)
= cosh x + log(x + \(\sqrt{x^2-1}\)) + C

Inter 2nd Year Maths 2B Integration Solutions Ex 6(a)

Question 13.
∫\(\frac{a^{x}-b^{x}}{a^{x}b^{x}}\) dx (a > 0, a ≠ 1 and b > 0, b ≠ 1) on R.
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(a) 15

Question 14.
∫sec² x cosec² x dx on I ⊂ R\ (nπ : n ∈ Z} ∪ { (2n + 1)\(\frac{\pi}{2}\) : n ∈ Z}).
Solution:
∫sec² x cosec² x dx
Inter 2nd Year Maths 2B Integration Solutions Ex 6(a) 16
= ∫\(\frac{1}{\cos^{2}x}\)dx + ∫\(\frac{1}{(\sin^{2}x}\)dx
= ∫sec² x dx + ∫cosec² x dx
= tan x – cot x + C

Question 15.
∫\(\frac{1+\cos^{2}x}{1+\cos2x}\) dx on I ⊂ R\{nπ :n ∈ Z}
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(a) 17

Question 16.
∫\(\sqrt{1-cos2x}\)dx on I ⊂ [2nπ, (2n + 1)π], n ∈ Z.
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(a) 18

Question 17.
∫\(\frac{1}{\cosh x+\sinh x}\) dx on R.
Solution:
∫\(\frac{1}{\cosh x+\sinh x}\) dx
= ∫\(\frac{\cosh x-\sinh x}{\cosh^{2}x-\sinh^{2}x}\) dx
= ∫(cosh x – sinh x) dx
= sinh x – cosh x + C

Inter 2nd Year Maths 2B Integration Solutions Ex 6(a)

Question 18.
∫\(\frac{1}{1+\cos x}\) dx on I ⊂ R \{(2n + 1)π : n ∈ Z}.
Solution:
Inter 2nd Year Maths 2B Integration Solutions Ex 6(a) 19
= ∫cosec² (x) dx – ∫cosec x cot x dx
= -cot x + cosec x + C

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