Cynthia Bell

2021-12-27

Evaluate the indefinite integral.
$\int \frac{3}{x}+\frac{2}{5{x}^{6}}dx$

censoratojk

Step 1
This question is based on indefinite integral.
General form,
Integral of ${x}^{n}$ is $\frac{{x}^{\left(x+1\right)}}{n+1}$
As for x = 1 formula violates because result comes is not defined.
Therefore, integral of 1/x is $\mathrm{ln}\left(x\right)$
Step 2
Given
$\int \left(\frac{3}{x}+\frac{2}{5{x}^{6}}\right)dx$
and $\int \frac{1}{x}dx=\mathrm{ln}\left(x\right)$
$=3\mathrm{ln}\left(x\right)+\frac{2}{5}\frac{{x}^{\left(-6+1\right)}}{\left(-6+1\right)}+C$
$=3\mathrm{ln}\left(x\right)+\frac{2}{5}\frac{{x}^{-5}}{-5}+C$
$=3\mathrm{ln}\left(x\right)-\frac{2}{25}{x}^{-5}+C$
Hence, Answer is $3\mathrm{ln}\left(x\right)-\frac{2}{25}{x}^{-5}+C$

Foreckije

$\int \left(\frac{3}{x}+\frac{2}{5\cdot {x}^{6}}\right)dx$
Lets

karton

Given:
$\int \frac{3}{x}+\frac{2}{5{x}^{6}}dx$
Use properties of integrals
$\int \frac{3}{x}dx+\int \frac{2}{5{x}^{6}}dx$
Evaluate the integrals
$3\mathrm{ln}\left(|x|\right)-\frac{2}{25{x}^{5}}$
$3\mathrm{ln}\left(|x|\right)-\frac{2}{25{x}^{5}}+C$