Find the following integrals. (i) ∫sinx/1+cos^2x dx, x ∈ [π/2,0]

Question

Find the following integrals.

(i) \(\int\limits_0^\frac{π}{2} \frac{sinx}{1+cos^2x}dx\)

(ii) \(\int\limits_0^1{xe^{x^2}}dx\)

(iii) \(\int\limits_0^{\frac{π}{2}} \sqrt{sinx}\,cos ^5 xdx\)

Answer 1

(i) I = \(\int\limits_0^\frac{π}{2} \frac{sinx}{1+cos^2x}dx\)

Put cosx = t ⇒ -sin xdx = dt

When x = 0 ⇒ t = cos0 = 1,

(ii) I = \(\int\limits_0^1{xe^{x^2}}dx\)

Put x² = t ⇒ 2xdx = dt

When x = 0 ⇒ t = 0,

x = 1 ⇒ t = 1

I = \(\frac{1}{2}\)\(\int\limits_0^1e^t\)dt =

\(\Big[e^t\Big]^t_0\)

= [e¹ – e⁰] = e – 1.

Put sin x = t ⇒ cos xdx = dt

When x = 0 ⇒ t = sin0 = 0,