Question

find the indefinite integral and check the result by differentiating.

$$ int ( 8 x + 10 ) ^ { 3 } ( 8 ) d x $$

$$ int ( 8 x + 10 ) ^ { 3 } ( 8 ) d x = $$

Explanation:

Step1: Use substitution method

Let \( u = 8x+10 \), then \( du=8dx \).
The integral \( \int(8x + 10)^{3}(8)dx=\int u^{3}du \).

Step2: Integrate \( u^{3} \)

Using the power - rule for integration \( \int x^{n}dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)), for \(n = 3\) and \(x = u\), we have \( \int u^{3}du=\frac{u^{4}}{4}+C\).

Step3: Substitute back \(u = 8x + 10\)

We get \( \frac{(8x + 10)^{4}}{4}+C\).

Step4: Check by differentiating

Differentiate \( y=\frac{(8x + 10)^{4}}{4}+C\) using the chain rule.
If \(y=\frac{u^{4}}{4}+C\) with \(u = 8x+10\), then \(y^\prime=\frac{4u^{3}}{4}\cdot\frac{du}{dx}\).
Since \(u = 8x + 10\), \(\frac{du}{dx}=8\).
So \(y^\prime=(8x + 10)^{3}\cdot8\), which is the integrand.

Answer:

\(\frac{(8x + 10)^{4}}{4}+C\)