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

To solve the integral \( \int 5 x^{4} (x^{5} + 10)^{3} d x \) using the substitution \( u = x^{5} + 10 \), first, we need to find \( du \): 1. Differentiate \( u \): \[ du = 5 x^{4} d x \] Now rewrite the integral in terms of \( u \): \[ \int 5 x^{4} (x^{5} + 10)^{3} d x = \int u^{3} du \] 2. Now, integrate: \[ \int u^{3} du = \frac{u^{4}}{4} + C \] 3. Substitute back \( u = x^{5} + 10 \): \[ \frac{(x^{5} + 10)^{4}}{4} + C \] Thus, the solution to the integral is: \[ \int 5 x^{4} (x^{5} + 10)^{3} d x = \frac{(x^{5} + 10)^{4}}{4} + C \] Now, let's check our answer by differentiating the result: 1. Differentiate \( \frac{(x^{5} + 10)^{4}}{4} + C \): \[ \frac{d}{dx} \left( \frac{(x^{5} + 10)^{4}}{4} \right) = \frac{1}{4} \cdot 4(x^{5} + 10)^{3} \cdot \frac{d}{dx}(x^{5} + 10) \] \[ = (x^{5} + 10)^{3} \cdot 5 x^{4} \] This matches the integrand \( 5 x^{4} (x^{5} + 10)^{3} \). Therefore, the original integral evaluates to: \[ \int 5 x^{4} (x^{5} + 10)^{3} d x = \frac{(x^{5} + 10)^{4}}{4} + C. \]