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Step-by-step Solution

Integral of $\int te^{\left(5t+\pi \right)}dt$

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Answer

$e^{\left(5t+\pi \right)}\left(-\frac{1}{25}+\frac{1}{5}t\right)+C_0$

Step-by-step explanation

Problem to solve:

$\int t e^{\left(5t+\pi \right)}dt$
1

Use the integration by parts theorem to calculate the integral $\int te^{\left(5t+\pi \right)}dt$, using the following formula

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
2

First, identify $u$ and calculate $du$

$\begin{matrix}\displaystyle{u=t}\\ \displaystyle{du=dt}\end{matrix}$

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Answer

$e^{\left(5t+\pi \right)}\left(-\frac{1}{25}+\frac{1}{5}t\right)+C_0$
$\int t e^{\left(5t+\pi \right)}dt$

Main topic:

Integration by substitution

Related formulas:

5. See formulas

Time to solve it:

~ 0.08 seconds