Harold Kessler

Answered

2021-12-30

Evaluate the indefinite integral.

$\int \frac{{\mathrm{sec}}^{2}\left(\sqrt{x}\right)dx}{\sqrt{x}}$

Answer & Explanation

vicki331g8

Expert

2021-12-31Added 37 answers

Step 1

We have to evaluate the integral:

$\int \frac{{\mathrm{sec}}^{2}\left(\sqrt{x}\right)dx}{\sqrt{x}}$

We will use substitution method since derivatives of one function is present in the integral.

So assuming,

$t=\sqrt{x}$

differentiating with respect to x,

$\frac{dt}{dx}=\frac{d\sqrt{x}}{dx}$

$\frac{dt}{dx}=\frac{1}{2\sqrt{x}}$

$2dt=\frac{dx}{\sqrt{x}}$

Substituting above values, we get

$\int \frac{{\mathrm{sec}}^{2}\left(\sqrt{x}\right)dx}{\sqrt{x}}=\int {\mathrm{sec}}^{2}\left(t\right)\left(2dt\right)$

$=2\int {\mathrm{sec}}^{2}\left(t\right)dt$

Step 2

Since we know that

$\int {\mathrm{sec}}^{2}xdx=\mathrm{tan}x+C$

therefore,

$2\int {\mathrm{sec}}^{2}\left(t\right)dt=2\mathrm{tan}\left(t\right)+C$

$=2\mathrm{tan}\left(\sqrt{x}\right)+C$

Where, C is an arbitrary constant.

Hence, value of given integral is$2\mathrm{tan}\left(\sqrt{x}\right)+C$ .

We have to evaluate the integral:

We will use substitution method since derivatives of one function is present in the integral.

So assuming,

differentiating with respect to x,

Substituting above values, we get

Step 2

Since we know that

therefore,

Where, C is an arbitrary constant.

Hence, value of given integral is

usumbiix

Expert

2022-01-01Added 33 answers

Substitution

This is the well-known tabular integral:

We substitute the already calculated integrals:

Reverse replacement

Solution:

karton

Expert

2022-01-04Added 439 answers

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