Use differentiation to determine whether the integral formula is correct.
-$$\int ( 5 x + 6 ) ^ { - 2 } d x = - \frac { ( 5 x + 6 ) ^ { - 1 } } { 5 } + C$$

Question

Quizplus · Accepted Answer

Let's differentiate the right-hand side of the given integral formula using the chain rule:

\begin{align*}
\frac{d}{dx}\left(-\frac{(5x+6)^{-1}}{5}+Cight) &= -\frac{-1}{5}(5x+6)^{-2}\cdot5 \
&= \frac{(5x+6)^{-2}}{5}
\end{align*}

Now let's simplify the integrand in the original integral:

\begin{align*}
\frac{d}{dx}\left((5x+6)^{-1}ight) &= -1(5x+6)^{-2}\cdot5 \
&= -\frac{(5x+6)^{-2}}{5}
\end{align*}

Thus, we see that the derivative of the right-hand side of the given formula matches the integrand, confirming that the formula is correct.

Use Differentiation to Determine Whether the Integral Formula Is Correct ∫(5x+6)−2dx=−(5x+6)−15+C\int ( 5 x + 6 ) ^ { - 2 } d x = - \frac { ( 5 x + 6 ) ^ { - 1 } } { 5 } + C∫(5x+6)−2dx=−5(5x+6)−1​+C

Use Differentiation to Determine Whether the Integral Formula Is Correct $\int ( 5 x + 6 ) ^ { - 2 } d x = - \frac { ( 5 x + 6 ) ^ { - 1 } } { 5 } + C$