Integrals — Practice

Question 1

Find the anti derivative of cos 3x.

Accepted Answer

Since d/dx[(1/3)sin3x]=cos3x, this is the antiderivative. Answer: (1/3)sin 3x + C.

Question 2

Integrate: 2x/(1+x^2).

Accepted Answer

Put t=1+x^2, dt=2xdx, so the integral becomes integral of dt/t=log|t|. Answer: log(1+x^2) + C.

Question 3

Integrate: 1/sqrt(x^2+2x+2).

Accepted Answer

Completing the square, x^2+2x+2=(x+1)^2+1, matching the standard log form. Answer: log|x+1+sqrt(x^2+2x+2)| + C.

Question 4

Integrate: x/[(x+1)(x+2)].

Accepted Answer

Decomposing gives -1/(x+1)+2/(x+2), since x=A(x+2)+B(x+1) gives A=-1, B=2 at the roots. Answer: -log|x+1| + 2log|x+2| + C.

Question 5

Integrate: x log x.

Accepted Answer

With u=logx, dv=xdx, v=x^2/2, integration by parts gives (x^2/2)logx - integral of x/2 dx. Answer: (x^2/2)log x - x^2/4 + C.

Question 6

Evaluate: integral from 0 to pi/4 of sin 2x dx.

Accepted Answer

The antiderivative is -(1/2)cos2x, since d/dx[-(1/2)cos2x]=sin2x. At pi/4, cos(pi/2)=0; at 0, cos0=1. Answer: 1/2.

Question 7

Evaluate: integral from -pi/2 to pi/2 of sin^7 x dx.

Accepted Answer

sin^7x is an odd function, since sin^7(-x)=-sin^7x, so by property P7 the integral over this symmetric interval vanishes with no computation needed. Answer: 0.

Question 8

Integrate: e^(2x) sin x.

Accepted Answer

Applying integration by parts twice produces an equation relating the integral I to itself: 5I=e^(2x)(2sinx-cosx), which is then solved for I. Answer: e^(2x)(2sin x - cos x)/5 + C.

Practice — Integrals

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