Practicing the Intermediate 2nd Year Maths 2B Textbook Solutions Inter 2nd Year Maths 2B Definite Integrals Solutions Exercise 7(b) will help students to clear their doubts quickly.

## Intermediate 2nd Year Maths 2B Definite Integrals Solutions Exercise 7(b)

I. Evaluate the following definite integrals.

Question 1.

\(\int_0^a(a^2x-x^3) d x\)

Solution:

Question 2.

\(\int_2^3 \frac{2 x}{1+x^2} d x\)

Solution:

I = [ln|1 + x²|]³_{2}

= ln 10 – ln 5

= ln(10/5)

= ln 2

Question 3.

\(\int_0^\pi \sqrt{2+2 \cos \theta} d \theta\)

Solution:

Question 4.

\(\int_0^\pi\sin^3x\cos^3xd x\)

Solution:

Question 5.

\(\int_0^2|1-x|d x\)

Solution:

Question 6.

\(\int_{-\pi / 2}^{\pi / 2} \frac{\cos x}{1+e^x} d x\)

Solution:

\(\int_{-\pi / 2}^{\pi / 2} \frac{\cos x dx}{1+e^x}\) ………….. (i)

cos x is even function

e^{x} is neither even nor odd.

Adding (i) and (ii) we get

Question 7.

\(\int_0^1\frac{dx}{\sqrt{3-2x}}\)

Solution:

3 – 2x = t²

-2dx = 2t dt

dx = -t dt

3 – (2.1) = t²

1 = t²

3 – 2.0 = t²

Question 8.

\(\int_0^a(\sqrt{a}-\sqrt{x})^2\)

Solution:

Question 9.

\(\int_0^{\pi / 4} \sec^4\theta d\theta\)

Solution:

Let ∫sec^{4} θ dθ = \(\int_0^{\pi / 4} \sec^2\theta(1+\tan^2\theta)d\theta\)

Put tan θ = y

sec² θ dθ = dy

θ = \(\frac{\pi}{4}\) ⇒ y= 1

θ = 0 ⇒ y = 0

I = \(\int_0^a\)(1 + y²)dy = [y + \(\frac{y^3}{3}\)]¹_{0}

= 1 + \(\frac{1}{3}=\frac{4}{3}\)

Question 10.

\(\int_0^3\frac{x}{\sqrt{x^2+16}}\)

Solution:

Question 11.

I = \(\int_0^1\)x.e^{-x²}dx

Solution:

I = \(\int_0^a\)x.e^{-x²}dx

⇒ -x² = t

⇒ -2x dx = dt

2x dx = -dt

x = 1 ⇒ t = 0

x = 0 ⇒ t = 1

I = \(\frac{1}{2}\)\(\int_0^a\)-e^{t }dt

= \(\frac{1}{2}\)[-e^{t}]^{-1}_{0}

= \(\frac{1}{2}\)[e^{0} – e^{-1}]

= \(\frac{1}{2}\)(1 – \(\frac{1}{e}\))

Question 12.

I = \(\int_1^5\frac{dx}{\sqrt{2x-1}}\)

Solution:

II. Evaluate the following integrals:

Question 1.

I = \(\int_0^4\frac{x^2}{1+x}\)

Solution:

Question 2.

\(\int_{-1}^2 \frac{x^2}{x^2+2}\)

Solution:

Question 3.

I = \(\int_0^1 \frac{x^2}{x^2+2}\)

Solution:

Question 4.

\(\int_0^{\pi / 2}\)x² sin x dx

Solution:

Question 5.

\(\int_0^4\)|2-x|dx

Solution:

Question 6.

\(\int_0^{\pi / 2}\frac{\sin^5 x}{\sin^5 x+\cos^5 x}\)

Solution:

Question 7.

\(\int_0^{\pi / 2}\frac{\sin^2 x-\cos^2 x}{\sin^3 x+\cos^3 x}\)dx

Solution:

Question 8.

Solution:

Question 9.

Solution:

Question 10.

Solution:

Question 11.

Solution:

Question 12.

Solution:

Question 13.

Solution:

Question 14.

Solution:

Question 15.

Solution:

III. Evaluate the following integrals :

Question 1.

\(\int_0^{\pi / 2}\frac{dx}{4+5\cos x}\)

Solution:

Question 2.

\(\int_a^b\sqrt{(x-a)(b-x)}\)dx

Solution:

Question 3.

Solution:

Question 4.

Solution:

Question 5.

Solution:

Question 6.

\(\int_0^a\)x(a – x)^{n}dx

Solution:

I = \(\int_0^a\)x(a – x)^{n}dx ………… (i)

\(\int_0^a\)f(x) dx = \(\int_0^a\)f(a – x)dx

I = \(\int_0^a\)(a – x).(x)^{n}dx ………… (ii)

I = \(\int_0^a\)ax^{n} dx – x^{n+1} dx

Question 7.

\(\int_0^2x\sqrt{2-x}\)dx

Solution:

I = \(\int_0^2x.\sqrt{2-x}\)dx

\(\int_0^2\)f(x)dx = \(\int_0^2\)f(a – x)dx

= \(\int_0^2\)(2 – x).√x dx

= \(\int_0^2\)((2√x – x√x)) dx

Question 8.

\(\int_0^{\pi}\)x sin³ x dx

Solution:

Question 9.

\(\int_0^{\pi}\frac{x}{1+\sin x}\)dx

Solution:

\(\int_0^{\pi}\frac{x}{1+\sin x}\)dx …………… (i)

Question 10.

Solution:

Question 11.

\(\int_0^1\frac{log(1+x)}{1+x^2}\)dx

Solution:

Put x = tan θ

dx = sec² θ dθ

x = 0 ⇒ θ = 0

x = x ⇒ θ = \(\frac{\pi}{4}\)

Question 12.

\(\int_0^{\pi}\frac{x \sin x}{1+\cos^2 x}\)dx

Solution:

Question 13.

\(\int_0^{\pi/2}\frac{\sin^2 x}{\cos x+\sin x}\)dx

Solution:

Question 14.

\(\int_0^{\pi}\frac{1}{3+2\cos x}\)dx

Solution:

Question 15.

\(\int_0^{\pi/4}\)log(1 + tan x)dx

Solution:

Question 16.

\(\int_{-1}^{3/2}\)|x sin πx| dx

Solution:

We know that |x. sin πx| = x . sin πx

where -1 ≤ x ≤ 1

and |x . sin πx| = – x.sin πx where 1 < x ≤ 3/2

Question 17.

\(\int_0^1sin^{-1}\frac{2x}{1+x^2}\)dx

Solution:

\(\int_0^1sin^{-1}\frac{2x}{1+x^2}\)dx

Put x = tan θ ⇒ dx = sec² θ dθ

x = 0 ⇒ θ = 0

x = 1 ⇒ θ π/4

Question 18.

\(\int_0^1\)x tan^{-1}x dx

Solution:

\(\int_0^1\)x. tan^{-1}x dx

Put x = tan θ ⇒ dx = sec² θ dθ

x = 0 ⇒ θ = 0;

Question 19.

\(\int_0^{\pi}\frac{x\sin x}{1+\cos^2 x}\)dx

Solution:

Question 20.

Suppose that f : R → R is continuous periodic function and T is the period of it. Let a ∈ R. Then prove that for any positive integer n,

Solution: