What is the integral of (45)/(1+25x^2) ?

The integral of (45)/(1+25x^2) is 9arctan(5x)+C

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integral of (45)/(1+25x^2)

\int \frac{45}{1 + 25 x ^{2}} d x

9 arctan (5 x) + C

Solution steps

\int \frac{45}{1 + 25 x ^{2}} d x

Take the constant out:

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

= 45 \cdot \int \frac{1}{1 + 25 x ^{2}} d x

Apply Integral Substitution

= 45 \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

= 45 \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)

= 45 \cdot \frac{1}{5} arctan (u)

Substitute back

u = 5 x

= 45 \cdot \frac{1}{5} arctan (5 x)

Simplify

45 \cdot \frac{1}{5} arctan (5 x) : 9 arctan (5 x)

= 9 arctan (5 x)

Add a constant to the solution

= 9 arctan (5 x) + C

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What is the integral of (45)/(1+25x^2) ?
The integral of (45)/(1+25x^2) is 9arctan(5x)+C