What is the integral of (10x)/(3x^2+15) ?

The integral of (10x)/(3x^2+15) is 5/3 ln|3x^2+15|+C

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integral of (10x)/(3x^2+15)

\int \frac{10 x}{3 x ^{2} + 15} d x

\frac{5}{3} ln 3 x^{2} + 15 + C

Solution steps

\int \frac{10 x}{3 x ^{2} + 15} d x

Take the constant out:

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

= 10 \cdot \int \frac{x}{3 x ^{2} + 15} d x

Apply u-substitution

: \int \frac{1}{6 u} d u

= 10 \cdot \int \frac{1}{6 u} d u

Take the constant out:

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

= 10 \cdot \frac{1}{6} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 10 \cdot \frac{1}{6} ln ∣ u ∣

Substitute back

u = 3 x^{2} + 15

= 10 \cdot \frac{1}{6} ln 3 x^{2} + 15

Simplify

10 \cdot \frac{1}{6} ln 3 x^{2} + 15 : \frac{5}{3} ln 3 x^{2} + 15

= \frac{5}{3} ln 3 x^{2} + 15

Add a constant to the solution

= \frac{5}{3} ln 3 x^{2} + 15 + C

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What is the integral of (10x)/(3x^2+15) ?
The integral of (10x)/(3x^2+15) is 5/3 ln|3x^2+15|+C