What is the integral of (15)/(25+x^2) ?

The integral of (15)/(25+x^2) is 3arctan(x/5)+C

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integral of (15)/(25+x^2)

\int \frac{15}{25 + x ^{2}} d x

3 arctan (\frac{x}{5}) + C

Solution steps

\int \frac{15}{25 + x ^{2}} d x

Take the constant out:

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

= 15 \cdot \int \frac{1}{25 + x ^{2}} d x

Apply Integral Substitution

= 15 \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

= 15 \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)

= 15 \cdot \frac{1}{5} arctan (u)

Substitute back

u = \frac{x}{5}

= 15 \cdot \frac{1}{5} arctan (\frac{x}{5})

Simplify

15 \cdot \frac{1}{5} arctan (\frac{x}{5}) : 3 arctan (\frac{x}{5})

= 3 arctan (\frac{x}{5})

Add a constant to the solution

= 3 arctan (\frac{x}{5}) + C

x \to - 2 lim (\frac{2}{x + 2})

\frac{d y}{d t} - 2 y = e^{2 t}

\int x^{5} (x^{6} - 10)^{4} d x

(x^{2} + 4) d y = (2 x - 8 y) d x

n \to 6 lim ((- 1)^{n} 7 n)

What is the integral of (15)/(25+x^2) ?
The integral of (15)/(25+x^2) is 3arctan(x/5)+C