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51.

 Find the integrals of the functions x sec-2 x

Answer»

∫xsec-2x

= x tan x -∫tan x + dx 

= x tan x + log |cos x| +C
Discussion
52.

Find the integrals of the functions  x sin x

Answer»

∫ x sin x . dx

= x(-cosx) + ∫1x cos xdx 

= – x cosx + sinx + C
Discussion
53.

Find the integrals of the functions  x2ex

Answer»

∫ x2ex .dx 

= x2ex -∫2x.ex dx 

= x2ex -2(xex -1 . ex) 

= x2ex – 2x ex + 2ex +C 

= ex (x2 – 2x + 2) + C
Discussion
54.

Consider the integral I = \(\int \frac{xsin^{-1}x}{\sqrt{1-x^2}}dx\)?What substitution can be given for simplifying the above integral? Express I in terms of the above substitution.Evaluate I

Answer»

1. Substitute sin-1 x = t.

2. We have, sin-1 x = t ⇒ x = sint
Differentiating w.r.t. x; we get,

\(\frac{1}{\sqrt{1-x^2}}dx = dt\)

∴ I = ∫t sin t dt.

3. I = ∫t sin t dt = t.(-cost) -∫(-cost)dt = -t cost + sint + c
= -sin-1 x. cos (sin-1 x) + sin(sin-1 x) + c
x – sin-1 x.cos(sin-1 x) + c.
Discussion
55.

Evaluate as limit of sum \(\int_2^3\frac{1}{x}dx\)

Answer»

\(\int_2^3\frac{1}{x}dx\) 

=\(\int_2^3logx\)

= log3 - log2

= log(\(\frac{3}{2}\))
Discussion
56.

∫ f (x→ -a,a)(x) dx = 0 if f is an _______ function.

Answer»

Answer is Odd
Discussion