evaluate the indefinite integral given below. ∫(−5x⁴ + 8x³)(−2x⁵ + 4x⁴)⁶ dx sorry, thats incorrect. try again?

Explanation:

Step1: Use substitution

Let $u = - 2x^{5}+4x^{4}$. Then $du=(-10x^{4} + 16x^{3})dx = 2(-5x^{4}+8x^{3})dx$, so $(-5x^{4}+8x^{3})dx=\frac{1}{2}du$.

Step2: Rewrite the integral

The original integral $\int(-5x^{4}+8x^{3})(-2x^{5}+4x^{4})^{6}dx$ becomes $\int u^{6}\cdot\frac{1}{2}du$.

Step3: Integrate with respect to u

Using the power - rule for integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$), we have $\frac{1}{2}\int u^{6}du=\frac{1}{2}\cdot\frac{u^{7}}{7}+C=\frac{u^{7}}{14}+C$.

Step4: Substitute back u

Substitute $u=-2x^{5}+4x^{4}$ back into the result, we get $\frac{(-2x^{5}+4x^{4})^{7}}{14}+C$.

Answer:

$\frac{(-2x^{5}+4x^{4})^{7}}{14}+C$