Evaluate the integral. int-2x^(2)ln(4x)dx Answer:

Identify integral: Identify the integral to be solved. We need to evaluate the integral of the function -2x^2ln(4x) with respect to x. I = ∫-2x^2ln(4x)dxApply integration by parts: Apply integration by parts. Integration by parts formula is ∫udv = uv - ∫vdu. Let u = ln(4x) and dv = -2x^2dx. Then we need to find du and v. du = (1/x)dx and v = ∫-2x^2dx.Calculate v: Calculate v. v = ∫-2x^2dx v = -2 * (1/3)x^3 v = -2/3x^3Apply integration by parts formula: Apply the integration by parts formula. I = uv - ∫vdu I = ln(4x) * (-2/3x^3) - ∫(-2/3x^3) * (1/x)dx I = -2/3x^3ln(4x) + 2/3∫x^2dxIntegrate remaining term: Integrate the remaining term. ∫x^2dx = (1/3)x^3Substitute integral: Substitute the integral into the equation. I = -2/3x^3ln(4x) + 2/3 * (1/3)x^3 I = -2/3x^3ln(4x) + 2/9x^3Add constant of integration: Add the constant of integration. I = -2/3x^3ln(4x) + 2/9x^3 + C

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Evaluate the integral.

int-2x^(2)ln(4x)dx
Answer:

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

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Calculus

Find indefinite integrals using the substitution and by parts

Full solution

Q. Evaluate the integral.

\newline

\int-2 x^{2} \ln (4 x) d x

\newline

Answer:

Identify integral: Identify the integral to be solved. $\newline$ We need to evaluate the integral of the function $-2x^2\ln(4x)$ with respect to $x$ . $\newline$ $I = \int -2x^2\ln(4x)\,dx$
Apply integration by parts: Apply integration by parts. Integration by parts formula is $\int u\,dv = uv - \int v\,du$ . Let $u = \ln(4x)$ and $dv = -2x^2\,dx$ . Then we need to find $du$ and $v$ . $du = (1/x)\,dx$ and $v = \int -2x^2\,dx$ .
Calculate $v$ : Calculate $v$ .
$v = \int -2x^2 \, dx$
$v = -2 \times \left(\frac{1}{3}\right)x^3$
$v = -\frac{2}{3}x^3$
Apply integration by parts formula: Apply the integration by parts formula. $\newline$ $I = uv - \int vdu$ $\newline$ $I = \ln(4x) \cdot \left(-\frac{2}{3}x^3\right) - \int\left(-\frac{2}{3}x^3\right) \cdot \left(\frac{1}{x}\right)dx$ $\newline$ $I = -\frac{2}{3}x^3\ln(4x) + \frac{2}{3}\int x^2dx$
Integrate remaining term: Integrate the remaining term. $\int x^2 \, dx = \frac{1}{3}x^3$
Substitute integral: Substitute the integral into the equation. $\newline$ $I = -\frac{2}{3}x^3\ln(4x) + \frac{2}{3} \times \frac{1}{3}x^3$ $\newline$ $I = -\frac{2}{3}x^3\ln(4x) + \frac{2}{9}x^3$
Add constant of integration: Add the constant of integration. $\newline$ $I = -\frac{2}{3}x^3\ln(4x) + \frac{2}{9}x^3 + C$