What is the integral of (10800x)/(9x^2+675) ?

The integral of (10800x)/(9x^2+675) is 600ln|9x^2+675|+C

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integral of (10800x)/(9x^2+675)

\int \frac{10800 x}{9 x ^{2} + 675} d x

600 ln 9 x^{2} + 675 + C

Solution steps

\int \frac{10800 x}{9 x ^{2} + 675} d x

Take the constant out:

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

= 10800 \cdot \int \frac{x}{9 x ^{2} + 675} d x

Apply u-substitution

: \int \frac{1}{18 u} d u

= 10800 \cdot \int \frac{1}{18 u} d u

Take the constant out:

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

= 10800 \cdot \frac{1}{18} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 10800 \cdot \frac{1}{18} ln ∣ u ∣

Substitute back

u = 9 x^{2} + 675

= 10800 \cdot \frac{1}{18} ln 9 x^{2} + 675

Simplify

10800 \cdot \frac{1}{18} ln 9 x^{2} + 675 : 600 ln 9 x^{2} + 675

= 600 ln 9 x^{2} + 675

Add a constant to the solution

= 600 ln 9 x^{2} + 675 + C

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What is the integral of (10800x)/(9x^2+675) ?
The integral of (10800x)/(9x^2+675) is 600ln|9x^2+675|+C