int((4*x-4)*(x^(2)-2*x-15)^(8))dx

Factor Quadratic Expression: Factor the quadratic expression inside the parentheses. We need to factor the quadratic expression (x^2 - 2x - 15) to simplify the integration process. Factoring the quadratic expression, we get: x^2 - 2x - 15 = (x - 5)(x + 3)Rewrite Integral: Rewrite the integral with the factored form. Now we can rewrite the integral as: ∫(4x - 4) * ((x - 5)(x + 3))^8 dxApply Distributive Property: Apply the distributive property to simplify the integrand. We can distribute the constant 4 into the parentheses: ∫(4x - 4) * ((x - 5)(x + 3))^8 dx = ∫(4x * (x - 5)(x + 3)^8 - 4 * (x - 5)(x + 3)^8) dxSplit Integral: Split the integral into two separate integrals. We can split the integral into two parts: ∫(4x * (x - 5)(x + 3)^8) dx - ∫(4 * (x - 5)(x + 3)^8) dxApply Power Rule: Apply the power rule for integration to both integrals. The power rule for integration states that ∫x^n dx = (x^(n+1))/(n+1) + C, where C is the constant of integration. However, because we have a product of terms and a composite function, we cannot directly apply the power rule. We need to use substitution or another integration technique.

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$\int\left((4 \cdot x-4) \cdot\left(x^{2}-2 \cdot x-15\right)^{8}\right) d x$

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Algebra 1

Multiplication with rational exponents

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\int\left((4 \cdot x-4) \cdot\left(x^{2}-2 \cdot x-15\right)^{8}\right) d x

Factor Quadratic Expression: Factor the quadratic expression inside the parentheses. $\newline$ We need to factor the quadratic expression $(x^2 - 2x - 15)$ to simplify the integration process. $\newline$ Factoring the quadratic expression, we get: $\newline$ $x^2 - 2x - 15 = (x - 5)(x + 3)$
Rewrite Integral: Rewrite the integral with the factored form. $\newline$ Now we can rewrite the integral as: $\newline$ $\int(4x - 4) \cdot ((x - 5)(x + 3))^8 \, dx$
Apply Distributive Property: Apply the distributive property to simplify the integrand. $\newline$ We can distribute the constant $4$ into the parentheses: $\newline$ $\int(4x - 4) \cdot ((x - 5)(x + 3))^8 \, dx = \int(4x \cdot (x - 5)(x + 3)^8 - 4 \cdot (x - 5)(x + 3)^8) \, dx$
Split Integral: Split the integral into two separate integrals. $\newline$ We can split the integral into two parts: $\newline$ $\int(4x * (x - 5)(x + 3)^8) \, dx - \int(4 * (x - 5)(x + 3)^8) \, dx$
Apply Power Rule: Apply the power rule for integration to both integrals. $\newline$ The power rule for integration states that $\int x^n \, dx = \frac{x^{(n+1)}}{n+1} + C$ , where $C$ is the constant of integration. However, because we have a product of terms and a composite function, we cannot directly apply the power rule. We need to use substitution or another integration technique.