Use differentiation to determine whether the integral formula is correct.
-$$\int 20 ( 5 x + 5 ) ^ { 3 } d x = ( 5 x + 5 ) ^ { 4 } + C$$

Question

Quizplus · Accepted Answer

We can use the power rule of differentiation to check if the integral formula is correct.
Taking the derivative of the right-hand side:
\begin{align*}
\frac{d}{d x} [(5 x + 5 ) ^ { 4 } + C] &= \frac{d}{d x} (5 x + 5) ^ { 4 } + \frac{d}{d x} C \
&= 4(5 x + 5) ^ 3 \frac{d}{d x} (5 x + 5) \
&= 20(5 x + 5) ^ 3
\end{align*}
Now we can compare this to the integrand on the left-hand side of the given formula:
\begin{align*}
\int 20 ( 5 x + 5 ) ^ { 3 } d x &= 5 ( 5 x + 5 ) ^ { 4 } + C_1 \
&= ( 5 x + 5 ) ^ { 4 } + C_2
\end{align*}
where $C_1$ and $C_2$ are constants of integration. We can see that the two expressions differ only by a constant, so the given integral formula is indeed correct. Thus, the answer is A.

Use Differentiation to Determine Whether the Integral Formula Is Correct ∫20(5x+5)3dx=(5x+5)4+C\int 20 ( 5 x + 5 ) ^ { 3 } d x = ( 5 x + 5 ) ^ { 4 } + C∫20(5x+5)3dx=(5x+5)4+C

Use Differentiation to Determine Whether the Integral Formula Is Correct $\int 20 ( 5 x + 5 ) ^ { 3 } d x = ( 5 x + 5 ) ^ { 4 } + C$