Use the substitution $$ u=x^{5}+10 $$ to find the following indefinite integral. Check your answer by differentiating. $$ \int 5 x^{4}\left(x^{5}+10\right)^{3} d x $$ $$ \int 5 x^{4}\left(x^{5}+10\right)^{3} d x=\square $$

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Let $$ u = x^5 + 10 $$. Then, differentiate with respect to $$ x $$:

$$
\frac{du}{dx} = 5x^4 \quad \Longrightarrow \quad du = 5x^4 \, dx.
$$

The given integral becomes:

$$
\int 5x^4 \left(x^5+10\right)^3 \, dx = \int u^3 \, du.
$$

Integrate $$ u^3 $$ with respect to $$ u $$:

$$
\int u^3 \, du = \frac{u^4}{4} + C.
$$

Substitute back $$ u = x^5 + 10 $$:

$$
\frac{(x^5+10)^4}{4} + C.
$$

Now, to check the answer, differentiate:

$$
\frac{d}{dx}\left[\frac{(x^5+10)^4}{4}\right] = \frac{4}{4}(x^5+10)^3 \cdot \frac{d}{dx}(x^5+10) = (x^5+10)^3 \cdot 5x^4 = 5x^4 (x^5+10)^3.
$$

This is exactly the original integrand.

Thus, the final answer is:

$$
\boxed{\frac{(x^5+10)^4}{4} + C.}
$$
$$ \int 5x^4(x^5+10)^3 \, dx = \frac{(x^5+10)^4}{4} + C $$

Use the substitution to find the following indefinite integral. Check your answer by differentiating.

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