What is the integral of e^{(-x)/3}cos(5x) ?

The integral of e^{(-x)/3}cos(5x) is (45e^{-x/3}sin(5x))/(226)-(3e^{-x/3}cos(5x))/(226)+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)/3}cos(5x)

\int e^{\frac{- x}{3}} cos (5 x) d x

\frac{45 e ^{- \frac{x}{3}} sin ( 5 x )}{226} - \frac{3 e ^{- \frac{x}{3}} cos ( 5 x )}{226} + C

Solution steps

\int e^{\frac{- x}{3}} cos (5 x) d x

Simplify

= \int e^{- \frac{x}{3}} cos (5 x) d x

Apply Integration By Parts

: \frac{1}{5} e^{- \frac{x}{3}} sin (5 x) - \int - \frac{1}{15} e^{- \frac{x}{3}} sin (5 x) d x

= \frac{1}{5} e^{- \frac{x}{3}} sin (5 x) - \int - \frac{1}{15} e^{- \frac{x}{3}} sin (5 x) d x

Take the constant out:

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

= \frac{1}{5} e^{- \frac{x}{3}} sin (5 x) - (- \frac{1}{15} \cdot \int e^{- \frac{x}{3}} sin (5 x) d x)

Apply Integration By Parts

: - \frac{1}{5} e^{- \frac{x}{3}} cos (5 x) - \int \frac{1}{15} e^{- \frac{x}{3}} cos (5 x) d x

= \frac{1}{5} e^{- \frac{x}{3}} sin (5 x) - (- \frac{1}{15} (- \frac{1}{5} e^{- \frac{x}{3}} cos (5 x) - \int \frac{1}{15} e^{- \frac{x}{3}} cos (5 x) d x))

Take the constant out:

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

= \frac{1}{5} e^{- \frac{x}{3}} sin (5 x) - (- \frac{1}{15} (- \frac{1}{5} e^{- \frac{x}{3}} cos (5 x) - \frac{1}{15} \cdot \int e^{- \frac{x}{3}} cos (5 x) d x))

Therefore

\int e^{- \frac{x}{3}} cos (5 x) d x = \frac{1}{5} e^{- \frac{x}{3}} sin (5 x) - (- \frac{1}{15} (- \frac{1}{5} e^{- \frac{x}{3}} cos (5 x) - \frac{1}{15} \cdot \int e^{- \frac{x}{3}} cos (5 x) d x))

Isolate

\int e^{- \frac{x}{3}} cos (5 x) d x

= \frac{45 e ^{- \frac{x}{3}} sin ( 5 x )}{226} - \frac{3 e ^{- \frac{x}{3}} cos ( 5 x )}{226}

Add a constant to the solution

= \frac{45 e ^{- \frac{x}{3}} sin ( 5 x )}{226} - \frac{3 e ^{- \frac{x}{3}} cos ( 5 x )}{226} + C

x \to 0 lim (\frac{11 sin ( 8 x )}{x})

7 y^{3} + 9 x^{8} = 5 y + 11 x, (1, 1)

x^{4} + 5 x^{2} - x

f (x) = 8 e^{e}

x^{2} e^{x} - 2 x e^{x} + 6 e^{x}

What is the integral of e^{(-x)/3}cos(5x) ?
The integral of e^{(-x)/3}cos(5x) is (45e^{-x/3}sin(5x))/(226)-(3e^{-x/3}cos(5x))/(226)+C