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

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

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integral of 5/(x^2+5)

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

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

Solution steps

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

Take the constant out:

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

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

Apply Integral Substitution

= 5 \cdot \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

= 5 \cdot \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)

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

Substitute back

u = \frac{x}{5}

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

Simplify

5 \cdot \frac{1}{5} arctan (\frac{x}{5}) : 5 arctan (\frac{x}{5})

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

Add a constant to the solution

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

n = 1 \sum \infty \frac{6}{n ^{5}}

\frac{d}{d x} (- x^{3} + 5 a x^{2} - 3 a^{2} x - b)

f (x) = e^{x - 2}

x^{2} 36 - x^{2}

x^{2} d y - 2 x y d x = x^{4} cos (2 x) d x

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