Computing Integrals: Problems 3

Problem : Compute x sin(x)dx.

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

Problem : Find x²e^xdx.

Letting f (x) = x² and g(x) = e^x, we have

x²e^xdx	=	x²e^x\|₀¹ = 2xe^xdx
	=	e - 2xe^xdx

Applying integration by parts again, we get

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

x²e^xdx = e - 2

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

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

Thus

I = - sinⁿ⁺¹(x)cos(x) + (n + 1)

sinⁿ(x)dx - I

I =

sinⁿ⁺¹(x)cos(x) +

sinⁿ(x)dx

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