What is the integral from 0 to pi of xsin(x)cos(nx) ?

The integral from 0 to pi of xsin(x)cos(nx) is (pi(-1)^{-n})/((n+1)(-n+1))

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral from 0 to pi of xsin(x)cos(nx)

\int_{0}^{π} x sin (x) cos (n x) d x

\frac{π ( - 1 ) ^{- n}}{( n + 1 ) ( - n + 1 )}

Solution steps

\int_{0}^{π} x sin (x) cos (n x) d x

Apply Integration By Parts

: [\frac{1}{2 ( n + 1 ) ( - n + 1 )} (x (- cos (x + n x) (- n + 1) - cos (x - n x) (n + 1)) - 2 \cdot \int \frac{1}{2} (- \frac{1}{1 + n} cos (x + n x) - \frac{1}{1 - n} cos (x - n x)) d x (n + 1) (- n + 1))]_{0}^{π}

= [\frac{1}{2 ( n + 1 ) ( - n + 1 )} (x (- cos (x + n x) (- n + 1) - cos (x - n x) (n + 1)) - 2 \cdot \int \frac{1}{2} (- \frac{1}{1 + n} cos (x + n x) - \frac{1}{1 - n} cos (x - n x)) d x (n + 1) (- n + 1))]_{0}^{π}

\int \frac{1}{2} (- \frac{1}{1 + n} cos (x + n x) - \frac{1}{1 - n} cos (x - n x)) d x = \frac{1}{2} (- \frac{1}{( 1 + n ) ^{2}} sin (x + n x) - \frac{1}{( 1 - n ) ^{2}} sin (x - n x))

= [\frac{1}{2 ( n + 1 ) ( - n + 1 )} (x (- cos (x + n x) (- n + 1) - cos (x - n x) (n + 1)) - 2 \cdot \frac{1}{2} (- \frac{1}{( 1 + n ) ^{2}} sin (x + n x) - \frac{1}{( 1 - n ) ^{2}} sin (x - n x)) (n + 1) (- n + 1))]_{0}^{π}

2 \cdot \frac{1}{2} (- \frac{1}{( 1 + n ) ^{2}} sin (x + n x) - \frac{1}{( 1 - n ) ^{2}} sin (x - n x)) (n + 1) (- n + 1) = (- \frac{1}{( 1 + n ) ^{2}} sin (x + n x) - \frac{1}{( 1 - n ) ^{2}} sin (x - n x)) (n + 1) (- n + 1)

= [\frac{1}{2 ( n + 1 ) ( - n + 1 )} (x (- cos (x + n x) (- n + 1) - cos (x - n x) (n + 1)) - (- \frac{1}{( 1 + n ) ^{2}} sin (x + n x) - \frac{1}{( 1 - n ) ^{2}} sin (x - n x)) (n + 1) (- n + 1))]_{0}^{π}

Compute the boundaries:

\frac{π ( - 1 ) ^{- n}}{( n + 1 ) ( - n + 1 )}

= \frac{π ( - 1 ) ^{- n}}{( n + 1 ) ( - n + 1 )}

\int_{0}^{l n (4)} 3 x^{2} e^{3 x} d x

\int_{- e}^{e} x^{3} e^{x^{6}} d x

\int_{- 1}^{\frac{1}{2}} (t + 1) (t^{2} + 4) d t

\int_{0}^{1} x cos (n x) d x

\int_{4}^{7} \frac{1}{( x - 3 ) ^{2}} d x

What is the integral from 0 to pi of xsin(x)cos(nx) ?
The integral from 0 to pi of xsin(x)cos(nx) is (pi(-1)^{-n})/((n+1)(-n+1))