Question

find the indefinite integral and check the result by differentiating.
int 2 x left( x ^ { 2 } + 6
ight) ^ { 5 } d x
int 2 x left( x ^ { 2 } + 6
ight) ^ { 5 } d x =

Explanation:

Step1: Use substitution

Let \(u = x^{2}+6\), then \(du=2x dx\).
The integral \(\int 2x(x^{2}+6)^{5}dx\) becomes \(\int u^{5}du\).

Step2: Integrate \(u^{5}\)

Using the power rule \(\int u^{n}du=\frac{u^{n + 1}}{n+1}+C\) (\(n
eq - 1\)), for \(n = 5\), we have \(\frac{u^{6}}{6}+C\).

Step3: Substitute back \(u=x^{2}+6\)

We get \(\frac{(x^{2}+6)^{6}}{6}+C\).

Step4: Check by differentiating

Differentiate \(y=\frac{(x^{2}+6)^{6}}{6}+C\) using the chain rule. Let \(t=x^{2}+6\), \(y=\frac{t^{6}}{6}+C\).
\(y^\prime=\frac{6t^{5}}{6}\cdot(2x)\) (by \(\frac{dy}{dx}=\frac{dy}{dt}\cdot\frac{dt}{dx}\)).
Substitute \(t = x^{2}+6\) back, \(y^\prime=2x(x^{2}+6)^{5}\).

Answer:

\(\frac{(x^{2}+6)^{6}}{6}+C\)