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:

Question 5.
∫$$\frac{1-\sqrt{x}}{x}$$dx on (0, ∞).
Solution:

Question 6.
∫($$1+\frac{2}{x}-\frac{3}{x^2}$$) dx on I⊂R\{0}
Solution:

Question 7.
∫(x + $$\frac{4}{1+x^2}$$)dx on R.
Solution:

Question 8.
∫(ex $$\frac{1}{x}-\frac{2}{\sqrt{x^2+1}}$$)dx on I⊂R\[-1, 1].
Solution:

Question 9.
∫($$\frac{1}{1-x^2}+\frac{1}{1+x^2}$$)dx on (-1, 1).
Solution:

Question 10.
∫($$\frac{1}{1-x^2}+\frac{2}{1+x^2}$$)dx on (-1, 1).
Solution:

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

Question 12.
∫$$\frac{\sin^{2}x}{1+\cos2x}$$ dx on I ⊂ R \{(2n ± 1)π : n ∈ Z}
Solution:

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:

Question 3.
∫($$\frac{\sqrt{x}+1}{x}$$)² dx on (0, ∞).
Solution:

Question 4.
∫($$\frac{(3x+1)^2}{2x}$$) dx, x ∈ I ⊂ R\ {0}.
Solution:

Question 5.
∫($$\frac{2x-1}{3\sqrt{x}}$$)² dx on (0, ∞).
Solution:

Question 6.
∫($$\frac{1}{\sqrt{x}}+\frac{2}{\sqrt{x^2-1}}-\frac{3}{2x^2}$$)² dx on (0, ∞).
Solution:

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:

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

Question 13.
∫$$\frac{a^{x}-b^{x}}{a^{x}b^{x}}$$ dx (a > 0, a ≠ 1 and b > 0, b ≠ 1) on R.
Solution:

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

= ∫$$\frac{1}{\cos^{2}x}$$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:

Question 16.
∫$$\sqrt{1-cos2x}$$dx on I ⊂ [2nπ, (2n + 1)π], n ∈ Z.
Solution:

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

Question 18.
∫$$\frac{1}{1+\cos x}$$ dx on I ⊂ R \{(2n + 1)π : n ∈ Z}.
Solution:

= ∫cosec² (x) dx – ∫cosec x cot x dx
= -cot x + cosec x + C