Math Snap

STEP 1

1. We are given an integral with a substitution provided.
2. The substitution is $u = x^9 - 2$.
3. We need to rewrite the integral in terms of $u$ and evaluate it.

STEP 2

1. Identify and perform the substitution.
2. Rewrite the integral in terms of $u$.
3. Evaluate the integral in terms of $u$.
4. Substitute back to express the result in terms of $x$.

STEP 3

Identify the substitution: $u = x^9 - 2$.
Calculate the derivative of $u$ with respect to $x$:
$$\frac{du}{dx} = 9x^8$$ This implies:
$$du = 9x^8 \, dx$$ Solve for $dx$:
$$dx = \frac{1}{9x^8} \, du$$

STEP 4

Substitute $u = x^9 - 2$ and $dx = \frac{1}{9x^8} \, du$ into the integral:
$$\int 10 x^8 (x^9 - 2)^{10} \, dx = \int 10 x^8 u^{10} \cdot \frac{1}{9x^8} \, du$$ Simplify the expression:
$$= \int \frac{10}{9} u^{10} \, du$$

STEP 5

Evaluate the integral in terms of $u$:
$$\int \frac{10}{9} u^{10} \, du = \frac{10}{9} \cdot \frac{1}{11} u^{11} + c$$ $$= \frac{10}{99} u^{11} + c$$

Substitute back $u = x^9 - 2$ to express the result in terms of $x$:
$$\frac{10}{99} (x^9 - 2)^{11} + c$$