Integral of 1/(x^7^(1/5))

\int\frac{1}{\sqrt[5]{x^{7}}}dx

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Answer

$-\frac{5}{2}\cdot\frac{1}{\sqrt[5]{x^{2}}}+C_0$

Step by step solution

Problem

$\int\frac{1}{\sqrt[5]{x^{7}}}dx$
1

Applying the power of a power property

$\int\frac{1}{\sqrt[5]{x^{7}}}dx$
2

Rewrite the exponent using the power rule $\frac{a^m}{a^n}=a^{m-n}$, where in this case $m=0$

$\int x^{-\frac{7}{5}}dx$
3

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

$-\frac{5}{2}x^{-\frac{2}{5}}$
4

Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number

$-\frac{5}{2}\cdot\frac{1}{\sqrt[5]{x^{2}}}$
5

Apply the formula: $a\frac{1}{x}$$=\frac{a}{x}$, where $a=-\frac{5}{2}$ and $x=\sqrt[5]{x^{2}}$

$\frac{-\frac{5}{2}}{\sqrt[5]{x^{2}}}$
6

Rewrite the exponent using the power rule $\frac{a^m}{a^n}=a^{m-n}$, where in this case $m=0$

$-\frac{5}{2}x^{-\frac{2}{5}}$
7

Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number

$-\frac{5}{2}\cdot\frac{1}{\sqrt[5]{x^{2}}}$
8

Add the constant of integration

$-\frac{5}{2}\cdot\frac{1}{\sqrt[5]{x^{2}}}+C_0$

Answer

$-\frac{5}{2}\cdot\frac{1}{\sqrt[5]{x^{2}}}+C_0$

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Problem Analysis

Main topic:

Integral calculus

Time to solve it:

0.25 seconds

Views:

129