Tiffany Russell

2021-12-26

Evaluate the integral.

$\int \frac{1}{{x}^{2}+25}dx$

Bob Huerta

Beginner2021-12-27Added 41 answers

Step 1

Consider the given integral:

$\int \frac{1}{{x}^{2}+25}dx$

Step 2

Now, substitute the following:

$x=5\mathrm{tan}\theta$

$dx=\left(5{\mathrm{sec}}^{2}\theta \right)d\theta$

Step 3

Then we will have:

$\int \frac{1}{{x}^{2}+25}dx=\int \frac{1}{25{\mathrm{tan}}^{2}\theta +25}5{\mathrm{sec}}^{2}\theta d\theta$

$=\int \frac{5{\mathrm{sec}}^{2}\theta}{25({\mathrm{tan}}^{2}\theta +1)}d\theta$

$=\int \frac{5{\mathrm{sec}}^{2}\theta}{25{\mathrm{sec}}^{2}\theta}d\theta$

$=\int \frac{1}{5}d\theta$

$=\frac{\theta}{5}+c$

Step 4

Now, back substitute the value of theta and get the value of required integral:

$\int \frac{1}{{x}^{2}+25}dx=\frac{1}{5}{\mathrm{tan}}^{-1}\left(\frac{x}{5}\right)+c$

Consider the given integral:

Step 2

Now, substitute the following:

Step 3

Then we will have:

Step 4

Now, back substitute the value of theta and get the value of required integral:

ol3i4c5s4hr

Beginner2021-12-28Added 48 answers

Given:

$\int \frac{1}{{x}^{2}+25}dx$

$=\int \frac{5}{25{u}^{2}+25}du$

Simplifying:

$=\frac{1}{5}\int \frac{1}{{u}^{2}+1}du$

$\int \frac{1}{{u}^{2}+1}du$

$=\mathrm{arctan}\left(u\right)$

$\frac{1}{5}\int \frac{1}{{u}^{2}+1}du$

$=\frac{\mathrm{arctan}\left(u\right)}{5}$

$=\frac{\mathrm{arctan}\left(\frac{x}{5}\right)}{5}$

$\int \frac{1}{{x}^{2}+25}dx$

Answer:

$=\frac{\mathrm{arctan}\left(\frac{x}{5}\right)}{5}+C$

Simplifying:

Answer:

karton

Expert2022-01-04Added 613 answers

Given:

Evaluate the integral

Calculate

Add C

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