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Chapter 7: Problem 13

Evaluate the following integrals. $$\int x^{2} \ln x^{3} d x$$

Short Answer

Expert verified
Question: Find the value of the integral $$\int x^2 \ln x^3 d x$$. Answer: The value of the integral is $$\frac{1}{3}x^3 \ln x^3 - \frac{1}{3}x^3 + C$$, where C is the constant of integration.

Step by step solution

01

Identify the functions u and dv

To solve the integral using integration by parts, we need to identify the two functions, u and dv. Choose u as the function that becomes simpler when differentiated and dv as the function that is suitable to be integrated. In this case, let: $$u = \ln{x^3}$$ and $$d v = x^2 dx$$
02

Find du and v

Now we need to find du and v. Differentiating u with respect to x, we have: $$\frac{d u}{d x}=\frac{d}{dx}(\ln{x^3})$$ Using the chain rule (derivative of the outer function multiplied by the derivative of the inner function), we get: $$\frac{d u}{d x}=3 x^2 \cdot \frac{1}{x^3}$$ Simplifying the expression: $$d u=3 {x^{-1}} d{x}$$ Now, integrate dv to find v: $$v = \int{x^2 dx}$$ Integration yields: $$v=\frac{1}{3}x^3$$
03

Apply integration by parts formula

Integration by parts formula states that: $$\int u\ d v = u v - \int v\ d u$$ Substituting the u, du, and v values we found: $$\int x^2 \ln x^3 d x = \left(\ln x^3\right)\left(\frac{1}{3}x^3\right) - \int \left(\frac{1}{3}x^3\right)\left(3 x^{-1} d x\right)$$
04

Simplify and solve the integral

Simplify the integral: $$\int x^2 \ln x^3 d x = \frac{1}{3}x^3 \ln x^3 - \int x^2 d x$$ Now, integrate x^2: $$\int x^2 \ln x^3 d x = \frac{1}{3}x^3 \ln x^3 - \frac{1}{3}x^3 + C$$ Here, C is the constant of integration. So, the solution to the given integral is: $$\int x^2 \ln x^3 d x = \frac{1}{3}x^3 \ln x^3 - \frac{1}{3}x^3 + C$$

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