What is the integral of (27)/(1+9x^2) ?

The integral of (27)/(1+9x^2) is 9arctan(3x)+C

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integral of (27)/(1+9x^2)

\int \frac{27}{1 + 9 x ^{2}} d x

9 arctan (3 x) + C

Solution steps

\int \frac{27}{1 + 9 x ^{2}} d x

Take the constant out:

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

= 27 \cdot \int \frac{1}{1 + 9 x ^{2}} d x

Apply Integral Substitution

= 27 \cdot \int \frac{1}{3 ( 1 + u ^{2} )} d u

Take the constant out:

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

= 27 \cdot \frac{1}{3} \cdot \int \frac{1}{1 + u ^{2}} d u

Use the common integral:

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

= 27 \cdot \frac{1}{3} arctan (u)

Substitute back

u = 3 x

= 27 \cdot \frac{1}{3} arctan (3 x)

Simplify

27 \cdot \frac{1}{3} arctan (3 x) : 9 arctan (3 x)

= 9 arctan (3 x)

Add a constant to the solution

= 9 arctan (3 x) + C

y^{^{''}} - 8 y^{^{'}} + 3 y = x e^{x}

\frac{d}{d x} (cos (h) (x))

\frac{d y}{d x} = (2^{4 x - y})

\frac{d}{d x} ((x^{3} - x) e^{x})

\frac{d}{d x} (x^{2} + x^{- 2})

What is the integral of (27)/(1+9x^2) ?
The integral of (27)/(1+9x^2) is 9arctan(3x)+C