Evaluate the following integrals.

$\int \frac{1}{x+\sqrt{x}}dx$

he298c
2021-10-20
Answered

Evaluate the following integrals.

$\int \frac{1}{x+\sqrt{x}}dx$

You can still ask an expert for help

toroztatG

Answered 2021-10-21
Author has **98** answers

Step 1

Consider the integral:

$\int \frac{1}{x+\sqrt{x}}dx$

Apply u−substitution:

$u=\sqrt{x}$

$\int \frac{1}{x+\sqrt{x}}dx=\int \frac{2}{u+1}du$

Take the constant out:

$\int \frac{2}{u+1}du=2\cdot \int \frac{1}{u+1}du$

Let v=u+1

$=2\cdot \int \frac{1}{v}dv$

$=2\mathrm{ln}\left|v\right|+C$

$=2\mathrm{ln}|u+1|+C$

$=2\mathrm{ln}|\sqrt{x}+1|+C$

Step 2

Hence:

$\int \frac{1}{x+\sqrt{x}}dx=2\mathrm{ln}|\sqrt{x}+1|+C$

Consider the integral:

Apply u−substitution:

Take the constant out:

Let v=u+1

Step 2

Hence:

asked 2021-11-23

Evaluate the integral.

${\int}_{0}^{1}(4x-9{x}^{2})dx$

asked 2021-10-21

Evaluate the indefinite integral.

$\int \mathrm{cos}0{\mathrm{sin}}^{6}0d0$

asked 2021-01-06

Evaluate the following integrals.

$\int \frac{dx}{{x}^{3}-{x}^{2}}$

asked 2022-05-13

Integrate $\frac{9\sqrt{{\mathrm{tan}}^{-1}(3x)}}{7(9{x}^{2}+1)}$ with respect to x.

asked 2021-08-07

Evaluate the integral.

${\int}_{\frac{\pi}{4}}^{\frac{3\pi}{4}}8{\mathrm{csc}}^{2}xdx$

asked 2021-11-01

Convert the indefinite integral into definate integral using the interval [0,1]

$\int ({x}^{5}+\frac{1}{5}{x}^{5}+\frac{1}{4}{x}^{4})dx$

asked 2022-01-26

I was trying to solve the integral $\int \mathrm{sin}\left(2x\right)\mathrm{sin}\left(4x\right)dx$ . Heres