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Problem 10

# Evaluate the given integral using the substitution (or method) indicated. $$\int(x-1)^{2} e^{(x-1)^{3}} d x ; u=(x-1)^{3}$$

Expert verified
The short version of the answer is: $$\int (x-1)^2 e^{(x-1)^3}\, dx = \frac{1}{3} e^{(x-1)^3} + C$$
See the step by step solution

## Step 1: Identify the given integral and substitution

We are given the integral to evaluate: $$\int (x-1)^2 e^{(x-1)^3} dx$$ The substitution function is given by: $$u = (x-1)^3$$

## Step 2: Find the derivative of the substitution function

Calculate $$du/dx$$ of the substitution $$u = (x-1)^3$$: $$\frac{du}{dx} = 3(x-1)^2$$ Now, calculate $$dx$$ by rearranging this equation: $$dx = \frac{du}{3(x-1)^2}$$

## Step 3: Rewrite the integral in terms of u

Using the substitution function, we can see that: $$(x-1)^2 = \sqrt{u}$$ Replace $$(x-1)^2$$ with $$\sqrt{u}$$ and $$dx$$ with $$\frac{du}{3\sqrt{u^2}}$$ in the integral: $$\int (x-1)^2 e^{(x-1)^3}\, dx = \int \sqrt{u} e^{u} \cdot \frac{du}{3\sqrt{u^2}}$$

## Step 4: Simplify the integral

Notice that we can simplify the expression in the integral: $$\sqrt{u} \cdot \frac{1}{3\sqrt{u^2}} = \frac{1}{3\sqrt{u}}$$ Now, we can rewrite the integral as: $$\int \frac{1}{3\sqrt{u}} e^u\, du$$

## Step 5: Evaluate the new integral

We can pull the constant 1/3 out of the integral and simply integrate $$e^u$$ with respect to $$u$$: $$\frac{1}{3} \int \frac{1}{\sqrt{u}} e^u\, du = \frac{1}{3} \int e^u\, du$$ Now, integrate: $$\frac{1}{3} \int e^u\, du = \frac{1}{3} e^u + C$$ Where $$C$$ is the constant of integration.

## Step 6: Substitute back the original variable x

Now, we substitute back our original substitution $$u = (x-1)^3$$ to find the resulting function of $$x$$: $$\frac{1}{3} e^u + C = \frac{1}{3} e^{(x-1)^3} + C$$ So, the final result is: $$\int (x-1)^2 e^{(x-1)^3}\, dx = \frac{1}{3} e^{(x-1)^3} + C$$

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