What is the integral of (24q)/(4q^2+7) ?

The integral of (24q)/(4q^2+7) is 3ln|4q^2+7|+C

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integral of (24q)/(4q^2+7)

\int \frac{24 q}{4 q ^{2} + 7} d q

3 ln 4 q^{2} + 7 + C

Solution steps

\int \frac{24 q}{4 q ^{2} + 7} d q

Take the constant out:

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

= 24 \cdot \int \frac{q}{4 q ^{2} + 7} d q

Apply u-substitution

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

= 24 \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

= 24 \cdot \frac{1}{8} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 24 \cdot \frac{1}{8} ln ∣ u ∣

Substitute back

u = 4 q^{2} + 7

= 24 \cdot \frac{1}{8} ln 4 q^{2} + 7

Simplify

24 \cdot \frac{1}{8} ln 4 q^{2} + 7 : 3 ln 4 q^{2} + 7

= 3 ln 4 q^{2} + 7

Add a constant to the solution

= 3 ln 4 q^{2} + 7 + C

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What is the integral of (24q)/(4q^2+7) ?
The integral of (24q)/(4q^2+7) is 3ln|4q^2+7|+C