Math Snap

STEP 1

1. We are given the integral $\int x^{11} e^{x^{12}} \, dx$.
2. We assume $u > 0$ when $\ln 4$ appears, although $\ln 4$ does not directly appear in this problem.
3. We will use substitution to solve the integral.

STEP 2

1. Identify a substitution to simplify the integral.
2. Apply the substitution to the integral.
3. Simplify the integral in terms of the new variable.
4. Integrate with respect to the new variable.
5. Substitute back to the original variable.

STEP 3

Identify a substitution. Let:
$$u = x^{12}$$ Then, differentiate $u$ with respect to $x$:
$$\frac{du}{dx} = 12x^{11}$$

STEP 4

Solve for $dx$ in terms of $du$ and $x$:
$$du = 12x^{11} \, dx$$ $$dx = \frac{1}{12x^{11}} \, du$$

STEP 5

Substitute $u = x^{12}$ and $dx = \frac{1}{12x^{11}} \, du$ into the integral:
$$\int x^{11} e^{x^{12}} \, dx = \int x^{11} e^{u} \cdot \frac{1}{12x^{11}} \, du$$ Simplify the expression:
$$= \frac{1}{12} \int e^{u} \, du$$

STEP 6

Integrate with respect to $u$:
$$\frac{1}{12} \int e^{u} \, du = \frac{1}{12} e^{u} + C$$ where $C$ is the constant of integration.

Substitute back $u = x^{12}$ into the integrated result:
$$\frac{1}{12} e^{u} + C = \frac{1}{12} e^{x^{12}} + C$$ The evaluated integral is:
$$\boxed{\frac{1}{12} e^{x^{12}} + C}$$