Question

first use the substitution method to find the indefinite integral. then find it without using substitution. check whether the answers are equivalent.
int 10 x ^ { 9 } left( x ^ { 10 }
ight) ^ { 9 } d x
use the substitution method to evaluate the integral.
int 10 x ^ { 9 } left( x ^ { 10 }
ight) ^ { 9 } d x =

Explanation:

Step1: Set substitution

Let \(u = x^{10}\), then \(du=10x^{9}dx\).

Step2: Substitute into integral

The integral \(\int 10x^{9}(x^{10})^{9}dx\) becomes \(\int u^{9}du\).
Using the power - rule for integration \(\int t^{n}dt=\frac{t^{n + 1}}{n+1}+C\) (\(n
eq - 1\)), for \(n = 9\) and \(t = u\), we have \(\frac{u^{10}}{10}+C\).

Step3: Substitute back

Substitute \(u = x^{10}\) back, we get \(\frac{(x^{10})^{10}}{10}+C=\frac{x^{100}}{10}+C\).

Answer:

\(\frac{x^{100}}{10}+C\)