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

The integral of x/(25+x^2) is 1/2 ln|25+x^2|+C

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integral of x/(25+x^2)

\int \frac{x}{25 + x ^{2}} d x

\frac{1}{2} ln 25 + x^{2} + C

Solution steps

\int \frac{x}{25 + x ^{2}} d x

Apply u-substitution

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

= \int \frac{1}{2 u} 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} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{2} ln ∣ u ∣

Substitute back

u = 25 + x^{2}

= \frac{1}{2} ln 25 + x^{2}

Add a constant to the solution

= \frac{1}{2} ln 25 + x^{2} + C

x \to 4 lim (e^{4} + 4)

x \to 5 lim (\frac{x}{x ^{2} - 25})

\int \frac{sin ^{2} ( x )}{cos ^{2} ( x )} d x

x + 1

f (x) = 6^{2 x} + 3^{x}

What is the integral of x/(25+x^2) ?
The integral of x/(25+x^2) is 1/2 ln|25+x^2|+C