What is the integral of 1/((5+x^2)) ?

The integral of 1/((5+x^2)) is 1/(sqrt(5))arctan(x/(sqrt(5)))+C

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integral of 1/((5+x^2))

\int \frac{1}{( 5 + x ^{2} )} d x

\frac{1}{5} arctan (\frac{x}{5}) + C

Solution steps

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

Apply Integral Substitution

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Use the common integral:

\int \frac{1}{1 + u ^{2}} d u = arctan (u)

= \frac{1}{5} arctan (u)

Substitute back

u = \frac{x}{5}

= \frac{1}{5} arctan (\frac{x}{5})

Add a constant to the solution

= \frac{1}{5} arctan (\frac{x}{5}) + C

n = 1 \sum \infty \frac{1}{5 n ^{2}}

f (x) = \frac{x ^{2} + 3}{x ^{2}}

y = cos^{2} (3 x - 1)

x \to 0 - lim (\frac{8}{5 x ^{\frac{1}{2}}})

(5.4) (7.7)

What is the integral of 1/((5+x^2)) ?
The integral of 1/((5+x^2)) is 1/(sqrt(5))arctan(x/(sqrt(5)))+C