Evaluate. (Be sure to check by differentiating!) \[ \int x^{21} e^{x^{22}} d x \] Determine a change of variables from $x$ to $u$. Choose the correct answer below. A. $u=x^{22}$ B. $u=e^{x}$ c. $u=x^{21} e^{x}$ D. $u=x^{21}$ Write the integral in terms of $u$. \[ \int x^{21} e^{x^{22}} d x=\int(\square) d u \] (Type an exact answer. Use parentheses to clearly denote the argument of each function.) Evaluate the integral. \[ \int x^{21} e^{x^{22}} d x= \] (Type an exact answer. Use parentheses to clearly denote the argument of each function.)

Step 1 ：First, we choose the correct change of variables. Looking at the integral, we see that the exponent of $e$ is $x^{22}$, so we choose $u=x^{22}$. This corresponds to option A.

Step 2 ：Next, we need to express $dx$ in terms of $du$. We differentiate $u=x^{22}$ with respect to $x$ to get $du=22x^{21}dx$.

Step 3 ：We solve this equation for $dx$ to get $dx=\frac{du}{22x^{21}}$.

Step 4 ：Now we substitute $u$ and $dx$ into the integral. We get $\int x^{21} e^{u} \frac{du}{22x^{21}}$.

Step 5 ：The $x^{21}$ terms cancel out, leaving us with $\frac{1}{22}\int e^{u} du$.

Step 6 ：The integral of $e^{u}$ with respect to $u$ is $e^{u}$, so we get $\frac{1}{22}e^{u}+C$ where $C$ is the constant of integration.

Step 7 ：Finally, we substitute $x^{22}$ back in for $u$ to get our final answer in terms of $x$. This gives us $\frac{1}{22}e^{x^{22}}+C$.

Step 8 ：To check our answer, we differentiate $\frac{1}{22}e^{x^{22}}+C$ with respect to $x$. Using the chain rule, we get $x^{21}e^{x^{22}}$, which is the integrand, confirming that our answer is correct.

Step 9 ：So, the solution to the integral $\int x^{21} e^{x^{22}} dx$ is $\boxed{\frac{1}{22}e^{x^{22}}+C}$.