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

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

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

integral

\frac{1}{( 1 + 2 x ) ^{2}}

- \frac{1}{2 ( 1 + 2 x )} + C

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply the Power Rule

: - \frac{1}{u}

= \frac{1}{2} (- \frac{1}{u})

Substitute back

u = 1 + 2 x

= \frac{1}{2} (- \frac{1}{1 + 2 x})

Simplify

\frac{1}{2} (- \frac{1}{1 + 2 x}) : - \frac{1}{2 ( 1 + 2 x )}

= - \frac{1}{2 ( 1 + 2 x )}

Add a constant to the solution

= - \frac{1}{2 ( 1 + 2 x )} + C

cos (x) - 1

\frac{( x - 2 )}{( 2 x ^{2} - 8 x - 1 )}

\frac{x}{x - 6}

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

(1 + x) cos (1 + x)

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