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

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

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integral of x/(2+x^2)

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

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

Solution steps

\int \frac{x}{2 + 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 = 2 + x^{2}

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

Add a constant to the solution

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

f^{^{'}} (x) = x^{\frac{2}{3}}

y = x^{4} + 4 e^{x}, (0, 4)

\int \frac{x + 3}{x ^{2} + 6 x} d x

\int 3 e^{x} - \frac{1}{x} d x

f (x) = x^{2} + 45, at x = 2

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