What is the integral of x/(x^4+25) ?

The integral of x/(x^4+25) is 1/10 arctan((x^2)/5)+C

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integral of x/(x^4+25)

\int \frac{x}{x ^{4} + 25} d x

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

Solution steps

\int \frac{x}{x ^{4} + 25} d x

Apply u-substitution

: \int \frac{1}{2 ( u ^{2} + 25 )} d u

= \int \frac{1}{2 ( u ^{2} + 25 )} 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} + 25} d u

Apply Integral Substitution

= \frac{1}{2} \cdot \int \frac{1}{5 ( v ^{2} + 1 )} d v

Take the constant out:

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

= \frac{1}{2} \cdot \frac{1}{5} \cdot \int \frac{1}{v ^{2} + 1} d v

Use the common integral:

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

= \frac{1}{2} \cdot \frac{1}{5} arctan (v)

Substitute back

= \frac{1}{2} \cdot \frac{1}{5} arctan (\frac{x ^{2}}{5})

Simplify

\frac{1}{2} \cdot \frac{1}{5} arctan (\frac{x ^{2}}{5}) : \frac{1}{10} arctan (\frac{x ^{2}}{5})

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

Add a constant to the solution

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

\int \frac{5}{x} d x

y^{2} + (x^{2} \frac{d y}{d x}) = (x y) \frac{d y}{d x}

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

\int (\frac{1}{ln ( x )}) d x

\int x^{2} (x - 5) 7 - x d x

What is the integral of x/(x^4+25) ?
The integral of x/(x^4+25) is 1/10 arctan((x^2)/5)+C