What is the integral of 1/((1+3y)^2) ?

The integral of 1/((1+3y)^2) is -1/(3(1+3y))+C

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integral of 1/((1+3y)^2)

\int \frac{1}{( 1 + 3 y ) ^{2}} d y

- \frac{1}{3 ( 1 + 3 y )} + C

Solution steps

\int \frac{1}{( 1 + 3 y ) ^{2}} d y

Apply u-substitution

: \int \frac{1}{3 u ^{2}} d u

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

Take the constant out:

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

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

Apply the Power Rule

: - \frac{1}{u}

= \frac{1}{3} (- \frac{1}{u})

Substitute back

u = 1 + 3 y

= \frac{1}{3} (- \frac{1}{1 + 3 y})

Simplify

\frac{1}{3} (- \frac{1}{1 + 3 y}) : - \frac{1}{3 ( 1 + 3 y )}

= - \frac{1}{3 ( 1 + 3 y )}

Add a constant to the solution

= - \frac{1}{3 ( 1 + 3 y )} + C

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What is the integral of 1/((1+3y)^2) ?
The integral of 1/((1+3y)^2) is -1/(3(1+3y))+C