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Evaluate the integral.

intx^(2)(4x-5)^(2)dx
Answer:

Evaluate the integral. $\newline$ $\int x^{2}(4 x-5)^{2} \mathrm{~d} x$ $\newline$ Answer:

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Evaluate definite integrals using the chain rule

Full solution

Q. Evaluate the integral.

\newline

\int x^{2}(4 x-5)^{2} \mathrm{~d} x

\newline

Answer:

Expand and Multiply: Let's start by expanding the integrand $(4x-5)^2$ and then multiplying it by $x^2$ . $(4x-5)^2 = (4x)^2 - 2\cdot(4x)\cdot(5) + (5)^2 = 16x^2 - 40x + 25$ Now, multiply this by $x^2$ : $x^2(16x^2 - 40x + 25) = 16x^4 - 40x^3 + 25x^2$
Integrate Terms Separately: Next, we integrate each term separately with respect to $x$ .
$\int 16x^4 \, dx = \frac{16}{5}x^5 + C_1$
$\int(-40x^3) \, dx = \frac{-40}{4}x^4 + C_2 = -10x^4 + C_2$
$\int 25x^2 \, dx = \frac{25}{3}x^3 + C_3$
Combine Integrals and Constants: Now, we combine the integrals and the constants of integration. $\newline$ $\int x^2(4x-5)^2 \, dx = \frac{16}{5}x^5 - 10x^4 + \frac{25}{3}x^3 + C$ $\newline$ Where $C$ is the combined constant of integration ( $C_1 + C_2 + C_3$ ).

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