What is the integral of (8x^2)/((64+x^2)^2) ?

The integral of (8x^2)/((64+x^2)^2) is 1/2 arctan(x/8)-(4x)/(64+x^2)+C

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integral of (8x^2)/((64+x^2)^2)

\int \frac{8 x ^{2}}{( 64 + x ^{2} ) ^{2}} d x

\frac{1}{2} arctan (\frac{x}{8}) - \frac{4 x}{64 + x ^{2}} + C

Solution steps

\int \frac{8 x ^{2}}{( 64 + x ^{2} ) ^{2}} d x

Take the constant out:

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

= 8 \cdot \int \frac{x ^{2}}{( 64 + x ^{2} ) ^{2}} d x

Apply Integral Substitution

= 8 \cdot \int \frac{u ^{2}}{8 ( u ^{2} + 1 ) ^{2}} d u

Take the constant out:

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

= 8 \cdot \frac{1}{8} \cdot \int \frac{u ^{2}}{( u ^{2} + 1 ) ^{2}} d u

Take the partial fraction of

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

= 8 \cdot \frac{1}{8} \cdot \int \frac{1}{u ^{2} + 1} - \frac{1}{( u ^{2} + 1 ) ^{2}} d u

Apply the Sum Rule:

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

= 8 \cdot \frac{1}{8} (\int \frac{1}{u ^{2} + 1} d u - \int \frac{1}{( u ^{2} + 1 ) ^{2}} d u)

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

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

= 8 \cdot \frac{1}{8} (arctan (u) - \frac{1}{2} (arctan (u) + \frac{u}{1 + u ^{2}}))

Substitute back

u = \frac{x}{8}

= 8 \cdot \frac{1}{8} (arctan (\frac{x}{8}) - \frac{1}{2} (arctan (\frac{x}{8}) + \frac{\frac{x}{8}}{1 + ( \frac{x}{8} ) ^{2}}))

Simplify

8 \cdot \frac{1}{8} (arctan (\frac{x}{8}) - \frac{1}{2} (arctan (\frac{x}{8}) + \frac{\frac{x}{8}}{1 + ( \frac{x}{8} ) ^{2}})) : \frac{1}{2} arctan (\frac{x}{8}) - \frac{4 x}{64 + x ^{2}}

= \frac{1}{2} arctan (\frac{x}{8}) - \frac{4 x}{64 + x ^{2}}

Add a constant to the solution

= \frac{1}{2} arctan (\frac{x}{8}) - \frac{4 x}{64 + x ^{2}} + C

(2.6) (4.12)

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\frac{d}{d x} (2 π x (cos (π x^{2})))

What is the integral of (8x^2)/((64+x^2)^2) ?
The integral of (8x^2)/((64+x^2)^2) is 1/2 arctan(x/8)-(4x)/(64+x^2)+C