Evaluate. (Assume $x > 0$ .) Check by differentiating.
$\int x^{12} \ln x d x$
$\int x^{12} \ln x d x =$

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Final Answer: The integral of $x^{12} \ln x d x$ is $\frac{x^{13} \ln x}{13} - \frac{x^{13}}{169}$ .

Steps

Step 1 ：Identify the parts of the integral that will be 'u' and 'dv'. In this case, let 'u' be $\ln x$ and 'dv' be $x^{12} d x$ .

Step 2 ：Find 'du' and 'v'. 'du' is the derivative of 'u' and 'v' is the integral of 'dv'. So, 'du' is $\frac{1}{x}$ and 'v' is $\frac{x^{13}}{13}$ .

Step 3 ：Apply the formula for integration by parts, $\int u d v = u v - \int v d u$ . This gives us $\frac{x^{13} \ln x}{13} - \frac{x^{13}}{169}$ .

Step 4 ：Check the result by differentiating it and see if we get back the original integrand. The derivative of $\frac{x^{13} \ln x}{13} - \frac{x^{13}}{169}$ is $x^{12} \ln x$ , which is the original integrand.

Step 5 ：Final Answer: The integral of $x^{12} \ln x d x$ is $\frac{x^{13} \ln x}{13} - \frac{x^{13}}{169}$ .

Evaluate. (Assume x>0.) Check by differentiating.∫x12ln⁡xdx∫x12ln⁡xdx=

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Evaluate. (Assume $x > 0$ .) Check by differentiating.
$\int x^{12} \ln x d x$
$\int x^{12} \ln x d x =$