Evaluate the integral. int x(3x+4)^(2)dx Answer:

Substitution Method: Let's use substitution to solve the integral. We will let u = 3x + 4, then du = 3dx. We need to express x in terms of u and dx in terms of du.Express x in terms of u: First, we solve for x in terms of u: u = 3x + 4 implies x = (u - 4)/3.Express dx in terms of du: Next, we express dx in terms of du: since du = 3dx, we have dx = du/3.Substitute x and dx: Now we substitute x and dx in the integral: ∫ x(3x + 4)^2 dx becomes ∫ ((u - 4)/3)(u^2)(du/3).Simplify the integral: Simplify the integral: ∫ ((u - 4)/3)(u^2)(du/3) = (1/9)∫ u^2(u - 4) du = (1/9)∫ (u^3 - 4u^2) du.Find the antiderivative: Find the antiderivative: (1/9)∫ (u^3 - 4u^2) du = (1/9)((u^4)/4 - (4u^3)/3) + C.Simplify the antiderivative: Simplify the antiderivative: (1/9)((u^4)/4 - (4u^3)/3) + C = (1/36)u^4 - (4/27)u^3 + C.Substitute back for u: Substitute back for u: (1/36)(3x + 4)^4 - (4/27)(3x + 4)^3 + C.

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Evaluate the integral.

int x(3x+4)^(2)dx
Answer:

Evaluate the integral. $\newline$ $\int x(3 x+4)^{2} \mathrm{~d} x$ $\newline$ Answer:

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Calculus

Evaluate definite integrals using the chain rule

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Q. Evaluate the integral.

\newline

\int x(3 x+4)^{2} \mathrm{~d} x

\newline

Answer:

Substitution Method: Let's use substitution to solve the integral. We will let $u = 3x + 4$ , then $du = 3dx$ . We need to express $x$ in terms of $u$ and $dx$ in terms of $du$ .
Express $x$ in terms of $u$ : First, we solve for $x$ in terms of $u$ : $u = 3x + 4$ implies $x = (u - 4)/3$ .
Express $dx$ in terms of $du$ : Next, we express $dx$ in terms of $du$ : since $du = 3dx$ , we have $dx = \frac{du}{3}$ .
Substitute $x$ and $dx$ : Now we substitute $x$ and $dx$ in the integral: $\int x(3x + 4)^2 dx$ becomes $\int \left(\frac{u - 4}{3}\right)(u^2)\left(\frac{du}{3}\right)$ .
Simplify the integral: Simplify the integral: \int \left(\frac{u - $4$}{$3$}\right)u^\(2\left(\frac{du}{ $3$ }\right) = \frac{ $1$ }{ $9$ }\int u^ $2$ (u - $4$ ) du = \frac{ $1$ }{ $9$ }\int (u^ $3$ - $4$ u^ $2$ ) du.
Find the antiderivative: Find the antiderivative: $(1/9)\int (u^3 - 4u^2) du = (1/9)((u^4)/4 - (4u^3)/3) + C$ .
Simplify the antiderivative: Simplify the antiderivative: (\frac{$1$}{$9$})\left(\frac{u^$4$}{$4$} - \frac{$4$u^$3$}{$3$}\right) + C = \left(\frac{$1$}{$36$}\right)u^\(4 - \left(\frac{ $4$ }{ $27$ }\right)u^ $3$ + C.
Substitute back for u: Substitute back for u: $(\frac{1}{36})(3x + 4)^4 - (\frac{4}{27})(3x + 4)^3 + C$ .