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

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

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

integral

\frac{1}{( 1 + 2 x ^{2} )}

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

Solution steps

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

Apply Integral Substitution

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

Substitute back

u = 2 x

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

Add a constant to the solution

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

e^{- 4} x

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

cosh^{2} (x)

1 - cot^{2} (x)

\frac{- 2}{x ^{3}}

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