Integral of we^w

\int w e^wdw

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Answer

$-e^w+we^w+C_0$

Step by step solution

Problem

$\int w e^wdw$
1

Use the integration by parts theorem to calculate the integral $\int we^wdw$, using the following formula

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

First, identify $u$ and calculate $du$

$\begin{matrix}\displaystyle{u=w}\\ \displaystyle{du=dw}\end{matrix}$
3

Now, identify $dv$ and calculate $v$

$\begin{matrix}\displaystyle{dv=e^wdw}\\ \displaystyle{\int dv=\int e^wdw}\end{matrix}$
4

Solve the integral

$v=\int e^wdw$
5

The integral of the exponential function is given by the following formula $\displaystyle \int a^xdx=\frac{a^x}{\ln(a)}$, where $a > 0$ and $a \neq 1$

$e^w$
6

Now replace the values of $u$, $du$ and $v$ in the last formula

$we^w-\int e^wdw$
7

The integral of the exponential function is given by the following formula $\displaystyle \int a^xdx=\frac{a^x}{\ln(a)}$, where $a > 0$ and $a \neq 1$

$we^w-e^w$
8

Factoring by $e^w$

$\left(w-1\right)e^w$
9

Multiplying polynomials $e^w$ and $w+-1$

$we^w-e^w$
10

Add the constant of integration

$-e^w+we^w+C_0$

Answer

$-e^w+we^w+C_0$

Problem Analysis

Main topic:

Integration by parts

Time to solve it:

0.25 seconds

Views:

67