What is the integral of 5/(36-4w^2) ?

The integral of 5/(36-4w^2) is 5/24 (ln| w/3+1|-ln| w/3-1|)+C

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integral of 5/(36-4w^2)

\int \frac{5}{36 - 4 w ^{2}} d w

\frac{5}{24} (ln \frac{w}{3} + 1 - ln \frac{w}{3} - 1) + C

Solution steps

\int \frac{5}{36 - 4 w ^{2}} d w

Take the constant out:

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

= 5 \cdot \int \frac{1}{36 - 4 w ^{2}} d w

Apply Integral Substitution

= 5 \cdot \int \frac{1}{12 ( - u ^{2} + 1 )} d u

Take the constant out:

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

= 5 \cdot \frac{1}{12} \cdot \int \frac{1}{- u ^{2} + 1} d u

Use the common integral:

\int \frac{1}{- u ^{2} + 1} d u = \frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}

= 5 \cdot \frac{1}{12} (\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2})

Substitute back

u = \frac{2}{6} w

= 5 \cdot \frac{1}{12} (\frac{ln \frac{2}{6} w + 1}{2} - \frac{ln \frac{2}{6} w - 1}{2})

Simplify

5 \cdot \frac{1}{12} (\frac{ln \frac{2}{6} w + 1}{2} - \frac{ln \frac{2}{6} w - 1}{2}) : \frac{5}{24} (ln \frac{w}{3} + 1 - ln \frac{w}{3} - 1)

= \frac{5}{24} (ln \frac{w}{3} + 1 - ln \frac{w}{3} - 1)

Add a constant to the solution

= \frac{5}{24} (ln \frac{w}{3} + 1 - ln \frac{w}{3} - 1) + C

What is the integral of 5/(36-4w^2) ?
The integral of 5/(36-4w^2) is 5/24 (ln| w/3+1|-ln| w/3-1|)+C