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

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

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integral of e^{3t}*t

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

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

Solution steps

\int e^{3 t} 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

\frac{d}{d x} (\frac{1}{( 1 + 2 x )})

\frac{d}{d x} ((- 16 x^{2} + 36)^{2})

\frac{d}{d x} (\frac{x}{1 - ln ( x - 3 )})

x \to 0 lim (\frac{4 ^{x} - 2 ^{x}}{x})

\frac{d}{d x} (a - b x^{2})

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