What is the integral of 1/(x+2x^3) ?

The integral of 1/(x+2x^3) is ln|x|-1/2 ln|2x^2+1|+C

Popular >

integral of 1/(x+2x^3)

\int \frac{1}{x + 2 x ^{3}} d x

ln ∣ x ∣ - \frac{1}{2} ln 2 x^{2} + 1 + C

Solution steps

\int \frac{1}{x + 2 x ^{3}} d x

Take the partial fraction of

\frac{1}{x + 2 x ^{3}} : \frac{1}{x} - \frac{2 x}{2 x ^{2} + 1}

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

= ln ∣ x ∣ - \frac{1}{2} ln 2 x^{2} + 1

Add a constant to the solution

= ln ∣ x ∣ - \frac{1}{2} ln 2 x^{2} + 1 + C

\int \frac{1}{2 x ^{2} x ^{2} - 16} d x

\int \frac{7}{w ^{2} + 3 w + 3} d w

\int 4 cot^{3} (x) csc (x) d x

\int 1 + 3 ln (x + 1) d x

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

What is the integral of 1/(x+2x^3) ?
The integral of 1/(x+2x^3) is ln|x|-1/2 ln|2x^2+1|+C