What is the integral of (3x)/(4x^2+5) ?

The integral of (3x)/(4x^2+5) is 3/8 ln|4x^2+5|+C

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integral of (3x)/(4x^2+5)

\int \frac{3 x}{4 x ^{2} + 5} d x

\frac{3}{8} ln 4 x^{2} + 5 + C

Solution steps

\int \frac{3 x}{4 x ^{2} + 5} d x

Take the constant out:

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

= 3 \cdot \int \frac{x}{4 x ^{2} + 5} d x

Apply u-substitution

: \int \frac{1}{8 u} d u

= 3 \cdot \int \frac{1}{8 u} d u

Take the constant out:

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

= 3 \cdot \frac{1}{8} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 3 \cdot \frac{1}{8} ln ∣ u ∣

Substitute back

u = 4 x^{2} + 5

= 3 \cdot \frac{1}{8} ln 4 x^{2} + 5

Simplify

3 \cdot \frac{1}{8} ln 4 x^{2} + 5 : \frac{3}{8} ln 4 x^{2} + 5

= \frac{3}{8} ln 4 x^{2} + 5

Add a constant to the solution

= \frac{3}{8} ln 4 x^{2} + 5 + C

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What is the integral of (3x)/(4x^2+5) ?
The integral of (3x)/(4x^2+5) is 3/8 ln|4x^2+5|+C