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 = \frac { x ^ { 100 } } { 10 } + c
use another method to evaluate the integral.
int 10 x ^ { 9 } left( x ^ { 10 }
ight) ^ { 9 } d x = square

Explanation:

Step1: Simplify the integrand

First, simplify \(10x^{9}(x^{10})^{9}\). Using the power - of - a - power rule \((a^{m})^{n}=a^{mn}\), we have \((x^{10})^{9}=x^{90}\). Then \(10x^{9}(x^{10})^{9}=10x^{9}\cdot x^{90}\). Using the product rule \(a^{m}\cdot a^{n}=a^{m + n}\), we get \(10x^{9}\cdot x^{90}=10x^{9 + 90}=10x^{99}\).

Step2: Integrate the simplified function

Now, find \(\int10x^{99}dx\). Using the power rule for integration \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)), where \(n = 99\) and the constant factor \(a = 10\).

$$
LATEXBLOCK0
$$

Answer:

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