What is the integral of (2x+3)^{-3} ?

The integral of (2x+3)^{-3} is -1/(4(2x+3)^2)+C

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integral of (2x+3)^{-3}

\int (2 x + 3)^{- 3} d x

- \frac{1}{4 ( 2 x + 3 ) ^{2}} + C

Solution steps

\int (2 x + 3)^{- 3} d x

Apply exponent rule:

a^{- b} = \frac{1}{a ^{b}}

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

Apply u-substitution

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

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

Apply the Power Rule

: - \frac{1}{2 u ^{2}}

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

Substitute back

u = 2 x + 3

= \frac{1}{2} (- \frac{1}{2 ( 2 x + 3 ) ^{2}})

Simplify

\frac{1}{2} (- \frac{1}{2 ( 2 x + 3 ) ^{2}}) : - \frac{1}{4 ( 2 x + 3 ) ^{2}}

= - \frac{1}{4 ( 2 x + 3 ) ^{2}}

Add a constant to the solution

= - \frac{1}{4 ( 2 x + 3 ) ^{2}} + C

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

\frac{d}{d x} (\frac{1}{1 - 4 x})

\frac{d}{d x} (\frac{x ^{2}}{3 x - 4})

x \to - 5 lim (f (x))

\int 2 x^{5} d x

What is the integral of (2x+3)^{-3} ?
The integral of (2x+3)^{-3} is -1/(4(2x+3)^2)+C