What is the integral of (2x)/(x^3-4x) ?

The integral of (2x)/(x^3-4x) is -1/2 ln| x/2+1|+1/2 ln| x/2-1|+C

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integral of (2x)/(x^3-4x)

\int \frac{2 x}{x ^{3} - 4 x} d x

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

Solution steps

\int \frac{2 x}{x ^{3} - 4 x} d x

Take the constant out:

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

= 2 \cdot \int \frac{x}{x ^{3} - 4 x} d x

Factor

\frac{x}{x ^{3} - 4 x}

= 2 \cdot \int \frac{x}{x ( x + 2 ) ( x - 2 )} d x

Multiply by the conjugate of

x + 2 : \frac{x ^{2} - 4}{x - 2}

= 2 \cdot \int \frac{x}{x \frac{x ^{2} - 4}{x - 2} ( x - 2 )} d x

Simplify

\frac{x}{x \frac{x ^{2} - 4}{x - 2} ( x - 2 )} : \frac{1}{x ^{2} - 4}

= 2 \cdot \int \frac{1}{x ^{2} - 4} d x

Apply Integral Substitution

= 2 \cdot \int \frac{1}{2 ( u ^{2} - 1 )} d u

Take the constant out:

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

= 2 \cdot \frac{1}{2} \cdot \int \frac{1}{u ^{2} - 1} d u

Factor

u^{2} - 1 : - (- u^{2} + 1)

= 2 \cdot \frac{1}{2} \cdot \int \frac{1}{- ( - u ^{2} + 1 )} d u

Take the constant out:

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

= 2 \cdot \frac{1}{2} (- \int \frac{1}{- u ^{2} + 1} d u)

Use the common integral:

\int \frac{1}{- u ^{2} + 1} d u = \frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}

= 2 \cdot \frac{1}{2} (- (\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}))

Substitute back

u = \frac{x}{2}

= 2 \cdot \frac{1}{2} (- (\frac{ln \frac{x}{2} + 1}{2} - \frac{ln \frac{x}{2} - 1}{2}))

Simplify

2 \cdot \frac{1}{2} (- (\frac{ln \frac{x}{2} + 1}{2} - \frac{ln \frac{x}{2} - 1}{2})) : - \frac{1}{2} ln \frac{x}{2} + 1 + \frac{1}{2} ln \frac{x}{2} - 1

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

Add a constant to the solution

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

sin (π x)

\int \frac{9 x ^{2} - 9}{x ^{3}} d x

\int x e^{x^{3}} d x

\int \frac{1}{( 25 + 4 x ^{2} ) ^{\frac{3}{2}}} d x

\int e^{- \frac{x}{3}} sin (2 x) d x

What is the integral of (2x)/(x^3-4x) ?
The integral of (2x)/(x^3-4x) is -1/2 ln| x/2+1|+1/2 ln| x/2-1|+C