What is the integral from 0 to pi of sin(mx)e^{nx} ?

The integral from 0 to pi of sin(mx)e^{nx} is (-e^{pin}mcos(pim)+ne^{pin}sin(pim)+m)/(m^2+n^2)

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integral from 0 to pi of sin(mx)e^{nx}

\int_{0}^{π} sin (m x) e^{n x} d x

\frac{- e ^{πn} m cos ( πm ) + n e ^{πn} sin ( πm ) + m}{m ^{2} + n ^{2}}

Solution steps

\int_{0}^{π} sin (m x) e^{n x} d x

Compute the indefinite integral:

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

Compute the boundaries:

\int_{0}^{π} sin (m x) e^{n x} d x = - \frac{m e ^{nπ} cos ( mπ )}{m ^{2} + n ^{2}} + \frac{n e ^{nπ} sin ( mπ )}{m ^{2} + n ^{2}} - (- \frac{m}{m ^{2} + n ^{2}})

= - \frac{m e ^{nπ} cos ( mπ )}{m ^{2} + n ^{2}} + \frac{n e ^{nπ} sin ( mπ )}{m ^{2} + n ^{2}} - (- \frac{m}{m ^{2} + n ^{2}})

Simplify

= \frac{- e ^{πn} m cos ( πm ) + n e ^{πn} sin ( πm ) + m}{m ^{2} + n ^{2}}

\int sin (\frac{x π}{2}) d x

\int \frac{( x - 1 ) ^{3}}{x ^{2}} d x

g (t) = 4 cos (4 t) - 9 sin (4 t) + 2 cos (10 t)

\frac{\partial}{\partial x} (r sin (t) sin (x))

\frac{\partial}{\partial x} (e^{- (x^{2} + y^{2} + 2 z^{2})})

What is the integral from 0 to pi of sin(mx)e^{nx} ?
The integral from 0 to pi of sin(mx)e^{nx} is (-e^{pin}mcos(pim)+ne^{pin}sin(pim)+m)/(m^2+n^2)