What is the integral of x/(sqrt(x^4+9)) ?

The integral of x/(sqrt(x^4+9)) is 1/2 ln((|x^2+sqrt(9+x^4)|)/3)+C

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integral of x/(sqrt(x^4+9))

\int \frac{x}{x ^{4} + 9} d x

\frac{1}{2} ln (\frac{x ^{2} + 9 + x ^{4}}{3}) + C

Solution steps

\int \frac{x}{x ^{4} + 9} d x

Apply u-substitution

= \int \frac{1}{2 u ^{2} + 9} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{1}{u ^{2} + 9} d u

Apply Trigonometric Substitution

= \frac{1}{2} \cdot \int sec (v) d v

Use the common integral:

\int sec (v) d v = ln ∣ tan (v) + sec (v) ∣

= \frac{1}{2} ln ∣ tan (v) + sec (v) ∣

Substitute back

= \frac{1}{2} ln tan (arctan (\frac{1}{3} x^{2})) + sec (arctan (\frac{1}{3} x^{2}))

Simplify

\frac{1}{2} ln tan (arctan (\frac{1}{3} x^{2})) + sec (arctan (\frac{1}{3} x^{2})) : \frac{1}{2} ln (\frac{x ^{2} + 9 + x ^{4}}{3})

= \frac{1}{2} ln (\frac{x ^{2} + 9 + x ^{4}}{3})

Add a constant to the solution

= \frac{1}{2} ln (\frac{x ^{2} + 9 + x ^{4}}{3}) + C

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

What is the integral of x/(sqrt(x^4+9)) ?
The integral of x/(sqrt(x^4+9)) is 1/2 ln((|x^2+sqrt(9+x^4)|)/3)+C