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 x ^{2} e ^{x} dx .
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 |
2xe ^{x} dx | = | 2xe ^{x}|_{0} ^{1} - 2e ^{x} dx | |
= | 2e - (2e ^{x}|_{0} ^{1}) | ||
= | 2 |
x ^{2} e ^{x} dx = e - 2 |
Problem : Express sin^{n+2}(x)dx in terms of sin^{n}(x)dx by integrating by parts twice.
Let I = sin^{n+2}(x)dx , the integral we are trying to compute. Integrating by parts gives
sin^{n+2}(x)dx | = | sin^{n+1}(x)sin(x)dx | |
= | sin^{n+1}(x)(- cos(x)) - (n + 1)sin^{n}(x)cos(x)(- cos(x))dx | ||
= | -sin^{n+1}(x)cos(x) + (n + 1)sin^{n}(x)cos^{2}(x)dx |
sin^{n}(x)cos^{2}(x)dx | = | sin^{n}(x)(1 - sin^{2}(x))dx | |
= sin^{n}(x)dx - sin^{n+2}(x)dx | |||
= sin^{n}(x)dx - I |
I = - sin^{n+1}(x)cos(x) + (n + 1) sin^{n}(x)dx - I , |
I = sin^{n+1}(x)cos(x) + sin^{n}(x)dx |