Evaluate the following integrals:

Question:

Evaluate the following integrals:

$\int\left(\frac{m}{x}+\frac{x}{m}+m^{x}+x^{m}+m x\right) d x$

Solution:

$\int\left(\frac{m}{x}+\frac{x}{m}+m^{x}+x^{m}+m x\right) d x$

By Splitting, we get,

$\Rightarrow \int \frac{\mathrm{m}}{\mathrm{x}} \mathrm{dx}+\int \frac{\mathrm{x}}{\mathrm{m}} \mathrm{dx}+\int \mathrm{x}^{\mathrm{m}} \mathrm{dx}+\int \mathrm{m}^{\mathrm{x}} \mathrm{dx}+\int \mathrm{mxdx}$

By using formula,

$\int \frac{1}{x} d x=\log x+c$

$\Rightarrow m l o g x+\frac{1}{m} \int x d x+\int x^{m} d x+\int m^{x} d x+\int m x d x$

By using the formula,

$\int x^{n} d x=\frac{x^{n+1}}{n+1}$

$\Rightarrow m l o g x+\frac{\frac{1}{m} x^{1+1}}{1+1}+\frac{x^{m+1}}{m+1}+\int m^{x} d x+\frac{m x^{1+1}}{1+1}$

By using the formula,

$\int a^{x} d x=\frac{a^{x}}{\log a}$

$\Rightarrow m l o g x+\frac{\frac{1}{m} x^{2}}{2}+\frac{x^{m+1}}{m+1}+\frac{m^{x}}{\operatorname{logm}}+\frac{m x^{2}}{2}+c$

Leave a comment

Close
faculty

Click here to get exam-ready with eSaral

For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.

Download Now