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

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

I. Evaluate the following integrals.

Question 1.

∫\(\frac{dx}{\sqrt{2x-3x^2+1}}\)

Solution:

Question 2.

∫\(\frac{\sin \theta}{\sqrt{2-\cos^2 \theta}}\)dθ

Solution:

Question 3.

∫\(\frac{\cos x}{\sin^2 x+4sin x+5}\)dx

Solution:

t = sin x ⇒ dt = cos x dx

I = ∫\(\frac{dt}{t^2+4t+5}\) = ∫\(\frac{dt}{(t+2)^2+1}\)

= tan^{-1}(t + 2) + C

= tan^{-1}(sin x + 2) + C

Question 4.

∫\(\frac{dx}{1+\cos^2 x}\)

Solution:

Question 5.

∫\(\frac{dx}{2\sin^2 x+3\cos^2 x}\)

Solution:

Question 6.

∫\(\frac{1}{1+\tan x}\)dx

Solution:

Question 7.

∫\(\frac{1}{1-\cot x}\)dx

Solution:

II. Evaluate the following integrals.

Question 1.

∫\(\sqrt{1+3x-x^2}\)dx

Solution:

Question 2.

∫\(\frac{9\cos x-\sin x}{4\sin x+5\cos x}\)dx

Solution:

Question 3.

∫\(\frac{2\cos x+3\sin x}{4\cos x+5\sin x}\)dx

Solution:

Let 2 cos c + 3 sin x = A(4 cos x + 5 sin x) + B(-4 sin x + 5 cos x)

Equating the co-efficient of sin x and cos x,

we get

4A + 5B = 2

5A – 4B = 3

Question 4.

∫\(\frac{1}{1+\sin x+\cos x}\)dx

Solution:

Question 5.

∫\(\frac{1}{3x^2+x+1}\)dx

Solution:

Question 6.

∫\(\frac{dx}{\sqrt{5-2x^2+4x}}\)

Solution:

III. Evaluate the following integrals.

Question 1.

∫\(\frac{x+1}{\sqrt{x^2-x+1}}\)

Solution:

Question 2.

∫(6x + 5)\(\sqrt{6-2x^2+x}\)dx

Solution:

Let 6x + 5 = A(1 – 4x) + B

Equating the constants

A + B = 5

B = 5 – A = 5 + \(\frac{3}{2}\) = \(\frac{13}{2}\)

∫(6x + 5)\(\sqrt{6-2x^2+x}\)dx

Question 3.

∫\(\frac{dx}{4+5\sin x}\)

Solution:

t = tan \(\frac{x}{2}\) ⇒ dt = sec² \(\frac{x}{2}\) . \(\frac{1}{2}\)dx

Question 4.

∫\(\frac{1}{2-3\cos 2x}\)dx

Solution:

t = tan x ⇒ dt = sec² x dx

= (1 + tan² x)dx

= (1 +t²)dx

dx = \(\frac{dt}{1+t^2}\)

Question 5.

∫x\(\sqrt{1+x-x^2}\)dx

Solution:

Let x = A(1 – 2x) + B

Equating the coefficients of x

1 = -2 A ⇒ A = –\(\frac{1}{2}\)

Equating the constants

0 = A + B ⇒ B = -A = \(\frac{1}{2}\)

∫x\(\sqrt{1+x-x^2}\)dx

= –\(\frac{1}{2}\)∫(1 – 2x)\(\sqrt{1+x-x^2}\)dx + \(\frac{1}{2}\)∫\(\sqrt{1+x-x^2}\)dx

Question 6.

∫\(\frac{dx}{(1+x)\sqrt{3+2x-x^2}}\)

Solution:

Question 7.

∫\(\frac{dx}{4\cos x+3\sin x}\)

Solution:

Let t = tan\(\frac{x}{2}\) so that dx = \(\frac{2dt}{1+t^2}\)

Question 8.

∫\(\frac{1}{\sin x+\sqrt{3} \cos x}\)dx

Solution:

Let t = tan \(\frac{x}{2}\) so that dx = \(\frac{2dt}{1+t^2}\)

sin x = \(\frac{2t}{1+t^2}\), cos x = \(\frac{1-t^2}{1+t^2}\)

Question 9.

∫\(\frac{dx}{5+4\cos 2x}\).

Solution:

Question 10.

∫\(\frac{2\sin x+3\cos x+4}{3\sin x+4\cos x+5}\)dx.

Solution:

Let 2 sin x + 3 cos x + 4

= A(3 sin x + 4 cos x + 5) + 3(3 cos x – 4 sin x) + C

Equating the co-efficient of

sin x, we get 3A – 4B = 2

cos x, we get 4A + 3B = 3

Equating the constants

4 = 5A + C

C = 4 – 5A = 4 – 5.\(\frac{18}{25}\) = \(\frac{2}{5}\)

Substituting in (1)

I = \(\frac{18}{25}\). x + \(\frac{1}{25}\) log|3 sin x + 4 cos x + 5| – \(\frac{4}{5\left(3+\tan \frac{x}{2}\right)}\) + C

Question 11.

∫\(\sqrt{\frac{5-x}{x-2}}\) dx on (2, 5).

Solution:

Let 5 – x = A. \(\frac{d}{dx}\)(7x – 10 – x²) + B

⇒ 5 – x = A(7 – 2x) + B

Equating coffs. of like terms

-2A = -1 ⇒ A = \(\frac{1}{2}\)

7A + B = 5

7(+\(\frac{1}{2}\)) + B = 5 ⇒ B = 5 – \(\frac{7}{2}\) = \(\frac{3}{2}\)

∴ 5 – x = \(\frac{1}{2}\)(7 – 2x) + \(\frac{3}{2}\)

Question 12.

∫\(\sqrt{\frac{1+x}{1-x}}\) dx on (-1, 1).

Solution:

Question 13.

∫\(\frac{dx}{(1 – x)\sqrt{3-2x+x^2}}\) on (-1, 3).

Solution:

Put 1 – x = \(\frac{1}{1}\) ⇒ 1 – \(\frac{1}{1}\) = x\(\frac{1}{1-x}\) = t

dx = \(\frac{1}{t^2}\)dt

3 – 2x – x² = 3 – 2(\(\frac{1-1}{1}\)) – (\(\frac{1-1}{1}\))²

Question 14.

∫\(\frac{dx}{(x + 2)\sqrt{x+1}}\) on (-1, ∞).

Solution:

Put = x + 1 = t² ⇒ dx = 2t dt and

x + 2 = 1 + t²

Question 15.

∫\(\frac{dx}{(2x + 3)\sqrt{x+2}}\) on I ⊂ (-2, ∞)\{\(\frac{-3}{2}\)}.

Solution:

Put x + 2 = t² ⇒ dx = 2t dt and

2x + 3 = 2(t² – 2) + 3 = 2t² – 1

Question 16.

∫\(\frac{1}{(1+\sqrt{x}) \sqrt{x-x^2}}\)dx on (0, 1).

Solution:

Question 17.

∫\(\frac{dx}{(x + 1)\sqrt{2x^2+3x+1}}\) on

Solution:

Question 18.

∫\(\sqrt{e^x-4}\) dx on [log_{e} 4, ∞]

Solution:

Put e^{x} – 4 = t² ⇒ e^{x} dx = 2t dt

Question 19.

∫\(\sqrt{1+\sec x}\) dx on [(2n – \(\frac{1}{2}\))π – (2n + \(\frac{1}{2}\))π], (n ∈ Z).

Solution:

Question 20.

∫\(\frac{dx}{1+x^4}\) on R.

Solution: