Computing Integrals

Problem : Compute x sin(x)dx .

Solution for Problem 1 >>

Letting f (x) = x and g(x) = sin(x) in the formula for integration by parts, we have

x sin(x)dx	=	- x cos(x) - (- cos(x))dx
	=	- x cos(x) + sin(x) + C

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Problem : Find x ² e ^x dx .

Solution for Problem 2 >>

Letting f (x) = x ² and g(x) = e ^x , we have

x 2 e x dx	=	x 2 e x|0 1 = 2xe x dx
	=	e - 2xe x dx

Applying integration by parts again, we get

2xe x dx	=	2xe x|0 1 - 2e x dx
	=	2e - (2e x|0 1)
	=	2

Substituting the value of this integral back in the first computation yields

x 2 e x dx = e - 2

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Problem : Express sinⁿ⁺²(x)dx in terms of sinⁿ(x)dx by integrating by parts twice.

Solution for Problem 3 >>

Let I = sinⁿ⁺²(x)dx , the integral we are trying to compute. Integrating by parts gives

sinn+2(x)dx	=	sinn+1(x)sin(x)dx
	=	sinn+1(x)(- cos(x)) - (n + 1)sinn(x)cos(x)(- cos(x))dx
	=	-sinn+1(x)cos(x) + (n + 1)sinn(x)cos2(x)dx

Integrating this new integral by parts, we have

sinn(x)cos2(x)dx	=	sinn(x)(1 - sin2(x))dx
		= sinn(x)dx - sinn+2(x)dx
		= sinn(x)dx - I

Thus

I = - sinn+1(x)cos(x) + (n + 1) sinn(x)dx - I ,

so

I = sinn+1(x)cos(x) + sinn(x)dx

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