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

The integral of x/(1-2x^2) is -1/4 ln|1-2x^2|+C

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

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

- \frac{1}{4} ln 1 - 2 x^{2} + C

Solution steps

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

Apply u-substitution

: \int - \frac{1}{4 u} d u

= \int - \frac{1}{4 u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \frac{1}{4} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - \frac{1}{4} ln ∣ u ∣

Substitute back

u = 1 - 2 x^{2}

= - \frac{1}{4} ln 1 - 2 x^{2}

Add a constant to the solution

= - \frac{1}{4} ln 1 - 2 x^{2} + C

\frac{d}{d x} (\frac{1}{1 + 24 e ^{- 0.28 x}})

\int_{2}^{4} x^{2} d x

\int_{1}^{4} (3 x^{3} + x + 1) d x

\frac{d}{d x} (\frac{3}{8 x ^{\frac{5}{2}}})

y^{^{''}} + 25 y = 3 sec (5 t)

What is the integral of x/(1-2x^2) ?
The integral of x/(1-2x^2) is -1/4 ln|1-2x^2|+C