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

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

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integral of (4x)/((x^2+4)^2)

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{1}{2 u ^{2}} d u

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

Take the constant out:

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

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

Apply the Power Rule

: - \frac{1}{u}

= 4 \cdot \frac{1}{2} (- \frac{1}{u})

Substitute back

u = x^{2} + 4

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

Simplify

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

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

Add a constant to the solution

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

\frac{\partial}{\partial x} (x e^{y} + y e^{x})

\int_{0}^{1} \frac{49}{x ^{5}} d x

7 x, at x = 4

\int 8 u^{\frac{9}{8}} d u

x = y^{2} - 2, (- 2, 2)

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