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)

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

\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

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

\int \frac{2 x + 4}{x ^{2} + 4 x + 8} d x

\int x + 2 d x

x d x + sin^{2} (x^{- 1} y) (y d x - x d y) = 0

- \frac{3}{2} x^{2} + 5 x + 5

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