What is the integral of (7x)/((x^2+8)^2+25) ?

The integral of (7x)/((x^2+8)^2+25) is 7/10 arctan(1/5 (x^2+8))+C

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integral of (7x)/((x^2+8)^2+25)

\int \frac{7 x}{( x ^{2} + 8 ) ^{2} + 25} d x

\frac{7}{10} arctan (\frac{1}{5} (x^{2} + 8)) + C

Solution steps

\int \frac{7 x}{( x ^{2} + 8 ) ^{2} + 25} d x

Take the constant out:

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

= 7 \cdot \int \frac{x}{( x ^{2} + 8 ) ^{2} + 25} d x

Expand

(x^{2} + 8)^{2} + 25 : x^{4} + 16 x^{2} + 89

= 7 \cdot \int \frac{x}{x ^{4} + 16 x ^{2} + 89} d x

Apply u-substitution

: \int \frac{1}{4 u ( 16 u + u + 89 )} d u

= 7 \cdot \int \frac{1}{4 u ( 16 u + u + 89 )} d u

Take the constant out:

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

= 7 \cdot \frac{1}{4} \cdot \int \frac{1}{u ( 16 u + u + 89 )} d u

Apply u-substitution

: \int \frac{2}{16 v + v ^{2} + 89} d v

= 7 \cdot \frac{1}{4} \cdot \int \frac{2}{16 v + v ^{2} + 89} d v

Take the constant out:

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

= 7 \cdot \frac{1}{4} \cdot 2 \cdot \int \frac{1}{16 v + v ^{2} + 89} d v

Complete the square

16 v + v^{2} + 89 : (v + 8)^{2} + 25

= 7 \cdot \frac{1}{4} \cdot 2 \cdot \int \frac{1}{( v + 8 ) ^{2} + 25} d v

Apply u-substitution

: \int \frac{1}{w ^{2} + 25} d w

= 7 \cdot \frac{1}{4} \cdot 2 \cdot \int \frac{1}{w ^{2} + 25} d w

Apply Integral Substitution

= 7 \cdot \frac{1}{4} \cdot 2 \cdot \int \frac{1}{5 ( t ^{2} + 1 )} d t

Take the constant out:

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

= 7 \cdot \frac{1}{4} \cdot 2 \cdot \frac{1}{5} \cdot \int \frac{1}{t ^{2} + 1} d t

Use the common integral:

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

= 7 \cdot \frac{1}{4} \cdot 2 \cdot \frac{1}{5} arctan (t)

Substitute back

= 7 \cdot \frac{1}{4} \cdot 2 \cdot \frac{1}{5} arctan (\frac{x ^{4} + 8}{5})

Simplify

7 \cdot \frac{1}{4} \cdot 2 \cdot \frac{1}{5} arctan (\frac{x ^{4} + 8}{5}) : \frac{7}{10} arctan (\frac{1}{5} (x^{2} + 8))

= \frac{7}{10} arctan (\frac{1}{5} (x^{2} + 8))

Add a constant to the solution

= \frac{7}{10} arctan (\frac{1}{5} (x^{2} + 8)) + C

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What is the integral of (7x)/((x^2+8)^2+25) ?
The integral of (7x)/((x^2+8)^2+25) is 7/10 arctan(1/5 (x^2+8))+C