What is the integral of 5/((3+4x^2)) ?

The integral of 5/((3+4x^2)) is 5/(2sqrt(3))arctan(2/(sqrt(3))x)+C

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integral of 5/((3+4x^2))

integral

\frac{5}{( 3 + 4 x ^{2} )}

\frac{5}{2 3} arctan (\frac{2}{3} x) + C

Solution steps

\int \frac{5}{3 + 4 x ^{2}} d x

Take the constant out:

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

= 5 \cdot \int \frac{1}{3 + 4 x ^{2}} d x

Apply Integral Substitution

= 5 \cdot \int \frac{1}{2 3 ( 1 + u ^{2} )} d u

Take the constant out:

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

= 5 \cdot \frac{1}{2 3} \cdot \int \frac{1}{1 + u ^{2}} d u

Use the common integral:

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

= 5 \cdot \frac{1}{2 3} arctan (u)

Substitute back

u = \frac{2}{3} x

= 5 \cdot \frac{1}{2 3} arctan (\frac{2}{3} x)

Simplify

5 \cdot \frac{1}{2 3} arctan (\frac{2}{3} x) : \frac{5}{2 3} arctan (\frac{2}{3} x)

= \frac{5}{2 3} arctan (\frac{2}{3} x)

Add a constant to the solution

= \frac{5}{2 3} arctan (\frac{2}{3} x) + C

What is the integral of 5/((3+4x^2)) ?
The integral of 5/((3+4x^2)) is 5/(2sqrt(3))arctan(2/(sqrt(3))x)+C