What is the integral of e^xsin(ax) ?

The integral of e^xsin(ax) is -(ae^xcos(ax))/(a^2+1)+(e^xsin(ax))/(a^2+1)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of e^xsin(ax)

\int e^{x} sin (a x) d x

- \frac{a e ^{x} cos ( a x )}{a ^{2} + 1} + \frac{e ^{x} sin ( a x )}{a ^{2} + 1} + C

Solution steps

\int e^{x} sin (a x) d x

Apply Integration By Parts

: - e^{x} \frac{1}{a} cos (a x) - \int - e^{x} \frac{1}{a} cos (a x) d x

= - e^{x} \frac{1}{a} cos (a x) - \int - e^{x} \frac{1}{a} cos (a x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - e^{x} \frac{1}{a} cos (a x) - (- \frac{1}{a} \cdot \int e^{x} cos (a x) d x)

Apply Integration By Parts

: e^{x} \frac{1}{a} sin (a x) - \int e^{x} \frac{1}{a} sin (a x) d x

= - e^{x} \frac{1}{a} cos (a x) - (- \frac{1}{a} (e^{x} \frac{1}{a} sin (a x) - \int e^{x} \frac{1}{a} sin (a x) d x))

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - e^{x} \frac{1}{a} cos (a x) - (- \frac{1}{a} (e^{x} \frac{1}{a} sin (a x) - \frac{1}{a} \cdot \int e^{x} sin (a x) d x))

Therefore

\int e^{x} sin (a x) d x = - e^{x} \frac{1}{a} cos (a x) - (- \frac{1}{a} (e^{x} \frac{1}{a} sin (a x) - \frac{1}{a} \cdot \int e^{x} sin (a x) d x))

Isolate

\int e^{x} sin (a x) d x

= - \frac{a e ^{x} cos ( a x )}{a ^{2} + 1} + \frac{e ^{x} sin ( a x )}{a ^{2} + 1}

Add a constant to the solution

= - \frac{a e ^{x} cos ( a x )}{a ^{2} + 1} + \frac{e ^{x} sin ( a x )}{a ^{2} + 1} + C

\int sin (x) cos (\frac{x}{2}) d x

\int (\frac{6 t - 8}{( t - 3 ) ^{3}}) d t

\int 2 x sin^{2} (x) + 2 x cos^{2} (x) d x

\int 3 x (sin^{2} (x) + cos^{2} (x)) d x

\int \frac{1}{x ^{2} - 6 x + 9} d x

What is the integral of e^xsin(ax) ?
The integral of e^xsin(ax) is -(ae^xcos(ax))/(a^2+1)+(e^xsin(ax))/(a^2+1)+C