What is the integral of 1/(4+25x^2) ?

The integral of 1/(4+25x^2) is 1/10 arctan(5/2 x)+C

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integral of 1/(4+25x^2)

\int \frac{1}{4 + 25 x ^{2}} d x

\frac{1}{10} arctan (\frac{5}{2} x) + C

Solution steps

\int \frac{1}{4 + 25 x ^{2}} d x

Apply Integral Substitution

= \int \frac{1}{10 ( 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}{10} \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}{10} arctan (u)

Substitute back

u = \frac{5}{2} x

= \frac{1}{10} arctan (\frac{5}{2} x)

Add a constant to the solution

= \frac{1}{10} arctan (\frac{5}{2} x) + C

\int \frac{x cos ( x )}{sin ^{3} ( x )} d x

\int \frac{3}{x ^{2} + 3 x} d x

\int t^{2} sin^{3} (t^{3}) cos^{2} (t^{3}) d t

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

tan^{2} (x)

What is the integral of 1/(4+25x^2) ?
The integral of 1/(4+25x^2) is 1/10 arctan(5/2 x)+C