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

The integral of 3e^{3t}t is e^{3t}t-1/3 e^{3t}+C

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integral of 3e^{3t}t

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

e^{3 t} t - \frac{1}{3} e^{3 t} + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

= 3 \cdot \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

= 3 \cdot \frac{1}{9} \cdot \int e^{u} u d u

Apply Integration By Parts

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

= 3 \cdot \frac{1}{9} (e^{u} u - \int e^{u} d u)

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

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

Substitute back

u = 3 t

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

Simplify

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

= e^{3 t} t - \frac{1}{3} e^{3 t}

Add a constant to the solution

= e^{3 t} t - \frac{1}{3} e^{3 t} + C

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