What is the integral of 1/(5+25x^2) ?

The integral of 1/(5+25x^2) is 1/(5sqrt(5))arctan(sqrt(5)x)+C

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integral of 1/(5+25x^2)

\int \frac{1}{5 + 25 x ^{2}} d x

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

Solution steps

\int \frac{1}{5 + 25 x ^{2}} d x

Apply Integral Substitution

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

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

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

Substitute back

u = \frac{5}{5} x

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

Simplify

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

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

Add a constant to the solution

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

\int \frac{y ^{3}}{( 2 + y ^{2} ) ^{\frac{5}{2}}} d y

\int \frac{x - 3}{( x ^{2} - 5 x + 1 )} d x

\int \frac{x}{1 + x x} d x

\int \frac{x ^{2} + 2}{( x + 1 ) ( x ^{2} + 4 )} d x

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

What is the integral of 1/(5+25x^2) ?
The integral of 1/(5+25x^2) is 1/(5sqrt(5))arctan(sqrt(5)x)+C