What is the integral of e^6xcos(e^3x) ?

The integral of e^6xcos(e^3x) is e^3xsin(e^3x)+cos(e^3x)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of e^6xcos(e^3x)

integral

e^{6} x cos (e^{3} x)

e^{3} x sin (e^{3} x) + cos (e^{3} x) + C

Solution steps

\int e^{6} x cos (e^{3} x) d x

Take the constant out:

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

= e^{6} \cdot \int x cos (e^{3} x) d x

Apply u-substitution

: \int \frac{u cos ( u )}{e ^{6}} d u

= e^{6} \cdot \int \frac{u cos ( u )}{e ^{6}} d u

Take the constant out:

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

= e^{6} \frac{1}{e ^{6}} \cdot \int u cos (u) d u

Apply Integration By Parts

: u sin (u) - \int sin (u) d u

= e^{6} \frac{1}{e ^{6}} (u sin (u) - \int sin (u) d u)

\int sin (u) d u = - cos (u)

= e^{6} \frac{1}{e ^{6}} (u sin (u) - (- cos (u)))

Substitute back

u = e^{3} x

= e^{6} \frac{1}{e ^{6}} (e^{3} x sin (e^{3} x) - (- cos (e^{3} x)))

Simplify

e^{6} \frac{1}{e ^{6}} (e^{3} x sin (e^{3} x) - (- cos (e^{3} x))) : e^{3} x sin (e^{3} x) + cos (e^{3} x)

= e^{3} x sin (e^{3} x) + cos (e^{3} x)

Add a constant to the solution

= e^{3} x sin (e^{3} x) + cos (e^{3} x) + C

2 x + 5

\frac{1}{( x ^{3} + a ^{3} )}

0^{13} x + 1 (x + 1)^{2}

(1 + x^{2})^{- 1}

\frac{e ^{x}}{( x + 2 )}

What is the integral of e^6xcos(e^3x) ?
The integral of e^6xcos(e^3x) is e^3xsin(e^3x)+cos(e^3x)+C