Get the answer to "Evaluate the integral. ∫1x2+25dx"

High School
Calculus and Analysis
Calculus 2
Applications of integrals

Tiffany Russell

Answered question

2021-12-26

Evaluate the integral.

\int \frac{1}{x^{2} + 25} d x

Answer & Explanation

Bob Huerta

Beginner2021-12-27Added 41 answers

Step 1
Consider the given integral:

\int \frac{1}{x^{2} + 25} d x

Step 2
Now, substitute the following:

x = 5 \tan θ

d x = (5 \sec^{2} θ) d θ

Step 3
Then we will have:

\int \frac{1}{x^{2} + 25} d x = \int \frac{1}{25 \tan^{2} θ + 25} 5 \sec^{2} θ d θ

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

= \int \frac{5 \sec^{2} θ}{25 \sec^{2} θ} d θ

= \int \frac{1}{5} d θ

= \frac{θ}{5} + c

Step 4
Now, back substitute the value of theta and get the value of required integral:

\int \frac{1}{x^{2} + 25} d x = \frac{1}{5} \tan^{- 1} (\frac{x}{5}) + c

ol3i4c5s4hr

Beginner2021-12-28Added 48 answers

Given:

\int \frac{1}{x^{2} + 25} d x

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

Simplifying:

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

\int \frac{1}{u^{2} + 1} d u

= \arctan (u)

\frac{1}{5} \int \frac{1}{u^{2} + 1} d u

= \frac{\arctan (u)}{5}

= \frac{\arctan (\frac{x}{5})}{5}

\int \frac{1}{x^{2} + 25} d x

Answer:

= \frac{\arctan (\frac{x}{5})}{5} + C

karton

Expert2022-01-04Added 613 answers

Given:
$\int \frac{1}{x^{2} + 25} d x$
Evaluate the integral
$\frac{1}{5} \times \arctan (\frac{x}{5})$
Calculate
$\frac{\arctan (\frac{x}{5})}{5}$
Add C
$\frac{\arctan (\frac{x}{5})}{5} + C$

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