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

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

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

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

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

Solution steps

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

Apply Integral Substitution

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

Use the common integral:

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

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

9 - x^{2}

3 x - 5

\frac{d}{d x} ([x + (x + sin^{2} (x))^{5}]^{4})

y^{^{''}} - 4 y^{^{'}} + 20 y = 0

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