What is the integral of (24)/(1+25r^2) ?

The integral of (24)/(1+25r^2) is 24/5 arctan(5r)+C

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integral of (24)/(1+25r^2)

\int \frac{24}{1 + 25 r ^{2}} d r

\frac{24}{5} arctan (5 r) + C

Solution steps

\int \frac{24}{1 + 25 r ^{2}} d r

Take the constant out:

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

= 24 \cdot \int \frac{1}{1 + 25 r ^{2}} d r

Apply Integral Substitution

= 24 \cdot \int \frac{1}{5 ( 1 + u ^{2} )} d u

Take the constant out:

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

= 24 \cdot \frac{1}{5} \cdot \int \frac{1}{1 + u ^{2}} d u

Use the common integral:

\int \frac{1}{1 + u ^{2}} d u = arctan (u)

= 24 \cdot \frac{1}{5} arctan (u)

Substitute back

u = 5 r

= 24 \cdot \frac{1}{5} arctan (5 r)

Simplify

24 \cdot \frac{1}{5} arctan (5 r) : \frac{24}{5} arctan (5 r)

= \frac{24}{5} arctan (5 r)

Add a constant to the solution

= \frac{24}{5} arctan (5 r) + C

What is the integral of (24)/(1+25r^2) ?
The integral of (24)/(1+25r^2) is 24/5 arctan(5r)+C