What is the integral of e^{ax}cos(mx) ?

The integral of e^{ax}cos(mx) is (me^{ax}sin(mx))/(m^2+a^2)+(ae^{ax}cos(mx))/(m^2+a^2)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of e^{ax}cos(mx)

\int e^{a x} cos (m x) d x

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

Solution steps

\int e^{a x} cos (m x) d x

Apply Integration By Parts

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

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

Take the constant out:

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

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

Simplify

a \frac{1}{m} : \frac{a}{m}

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Simplify

- a \frac{1}{m} : - \frac{a}{m}

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

Therefore

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

Isolate

\int e^{a x} cos (m x) d x

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

Add a constant to the solution

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

\int \frac{x ^{2} + 1}{x ^{2} + 1} d x

\int \frac{sec ^{3} ( 2 x )}{tan ( 2 x )} d x

\int x sin (7 x^{2}) d x

\int (e^{- x} + 1)^{2} d x

\int x (x^{2} + 10)^{9} d x

What is the integral of e^{ax}cos(mx) ?
The integral of e^{ax}cos(mx) is (me^{ax}sin(mx))/(m^2+a^2)+(ae^{ax}cos(mx))/(m^2+a^2)+C