What is the integral of e^{-x}cos(pix) ?

The integral of e^{-x}cos(pix) is (pie^{-x}sin(pix))/(pi^2+1)-(e^{-x}cos(pix))/(pi^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^{-x}cos(pix)

\int e^{- x} cos (π x) d x

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

Solution steps

\int e^{- x} cos (π x) d x

Apply Integration By Parts

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

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Therefore

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

Isolate

\int e^{- x} cos (π x) d x

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

Add a constant to the solution

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

\int 0.4 x^{2} + 1000 d x

\int (\frac{2}{6 x}) d x

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

\int 10 - 2 t d t

\int cos (8 θ) d θ

What is the integral of e^{-x}cos(pix) ?
The integral of e^{-x}cos(pix) is (pie^{-x}sin(pix))/(pi^2+1)-(e^{-x}cos(pix))/(pi^2+1)+C