What is the integral of t*e^{3t} ?

The integral of t*e^{3t} is 1/9 (e^{3t}*3t-e^{3t})+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of t*e^{3t}

\int t \cdot e^{3 t} d t

\frac{1}{9} (e^{3 t} \cdot 3 t - e^{3 t}) + C

Solution steps

\int t e^{3 t} d t

Apply u-substitution

: \int \frac{e ^{u} u}{9} d u

= \int \frac{e ^{u} u}{9} d u

Take the constant out:

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

= \frac{1}{9} \cdot \int e^{u} u d u

Apply Integration By Parts

: e^{u} u - \int e^{u} d u

= \frac{1}{9} (e^{u} u - \int e^{u} d u)

\int e^{u} d u = e^{u}

= \frac{1}{9} (e^{u} u - e^{u})

Substitute back

u = 3 t

= \frac{1}{9} (e^{3 t} \cdot 3 t - e^{3 t})

Add a constant to the solution

= \frac{1}{9} (e^{3 t} \cdot 3 t - e^{3 t}) + C

\int 4 x^{2} - 12 x + 5 d x

\int \frac{sin ^{3} ( θ )}{1 + cos ( θ )} d θ

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

\int (3 x - 2) (x + 1)^{3} d x

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

What is the integral of t*e^{3t} ?
The integral of t*e^{3t} is 1/9 (e^{3t}*3t-e^{3t})+C