What is the integral of (8r)/(4r^2-5) ?

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

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integral of (8r)/(4r^2-5)

\int \frac{8 r}{4 r ^{2} - 5} d r

ln 4 r^{2} - 5 + C

Solution steps

\int \frac{8 r}{4 r ^{2} - 5} d r

Take the constant out:

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

= 8 \cdot \int \frac{r}{4 r ^{2} - 5} d r

Apply u-substitution

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

= 8 \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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 8 \cdot \frac{1}{8} ln ∣ u ∣

Substitute back

u = 4 r^{2} - 5

= 8 \cdot \frac{1}{8} ln 4 r^{2} - 5

Simplify

8 \cdot \frac{1}{8} ln 4 r^{2} - 5 : ln 4 r^{2} - 5

= ln 4 r^{2} - 5

Add a constant to the solution

= ln 4 r^{2} - 5 + C

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