Questions: Evaluate. (Be sure to check by differentiating!) ∫ x^12 e^(x^13) dx Determine a change of variables from x to u. Choose the correct option: A. u=e^x B. u=x^12 e^x C. u=x^12 D. u=x^13

Solution Steps

To solve the integral \(\int x^{12} e^{x^{13}} dx\), we can use the method of substitution. We need to choose a substitution that simplifies the integral. By examining the options, we notice that the derivative of \(x^{13}\) is \(13x^{12}\), which matches the \(x^{12}\) term in the integral. Therefore, the substitution \(u = x^{13}\) is appropriate. This will transform the integral into a simpler form in terms of \(u\).

1. Choose the substitution \(u = x^{13}\).
2. Compute the differential \(du = 13x^{12} dx\).
3. Solve for \(dx\) in terms of \(du\) and substitute into the integral.
4. Integrate with respect to \(u\).
5. Substitute back to express the result in terms of \(x\).

We start with the integral

\[ \int x^{12} e^{x^{13}} dx. \]

To simplify this integral, we choose the substitution

\[ u = x^{13}. \]

Next, we compute the differential of \(u\):

\[ du = 13x^{12} dx \implies dx = \frac{du}{13x^{12}}. \]

Substituting \(u\) and \(dx\) into the integral, we have:

\[ \int x^{12} e^{x^{13}} dx = \int x^{12} e^{u} \cdot \frac{du}{13x^{12}} = \frac{1}{13} \int e^{u} du. \]

The integral of \(e^{u}\) is

\[ e^{u} + C. \]

Thus, we have:

\[ \frac{1}{13} \int e^{u} du = \frac{1}{13} e^{u} + C. \]

Now, we substitute back \(u = x^{13}\):

\[ \frac{1}{13} e^{x^{13}} + C. \]

Final Answer