Integrate x^10+e^(5x)

\int\left(x^{10}+e^{5x}\right)dx

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Answer

$\frac{1}{5}e^{5x}+\frac{x^{11}}{11}+C_0$

Step by step solution

Problem

$\int\left(x^{10}+e^{5x}\right)dx$
1

The integral of a sum of two or more functions is equal to the sum of their integrals

$\int e^{5x}dx+\int x^{10}dx$
2

Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function

$\int e^{5x}dx+\frac{x^{11}}{11}$
3

Solve the integral $\int e^{5x}dx$ applying u-substitution. Let $u$ and $du$ be

$\begin{matrix}u=5x \\ du=5dx\end{matrix}$
4

Isolate $dx$ in the previous equation

$\frac{du}{5}=dx$
5

Substituting $u$ and $dx$ in the integral

$\int\frac{e^u}{5}du+\frac{x^{11}}{11}$
6

Taking the constant out of the integral

$\frac{1}{5}\int e^udu+\frac{x^{11}}{11}$
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 $a \neq 1$

$\frac{1}{5}e^u+\frac{x^{11}}{11}$
8

Substitute $u$ back for it's value, $5x$

$\frac{1}{5}e^{5x}+\frac{x^{11}}{11}$
9

Add the constant of integration

$\frac{1}{5}e^{5x}+\frac{x^{11}}{11}+C_0$

Answer

$\frac{1}{5}e^{5x}+\frac{x^{11}}{11}+C_0$

Problem Analysis

Main topic:

Integration by substitution

Time to solve it:

0.23 seconds

Views:

122