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

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

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

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

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

Solution steps

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

Apply Integral Substitution

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

Use the common integral:

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

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

Substitute back

u = \frac{r}{5}

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

Simplify

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

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

Add a constant to the solution

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

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