Math Snap

STEP 1

1. The given problem involves evaluating a double integral.
2. The integrals are with respect to different variables, suggesting a potential separation of variables.
3. We assume $y$ is treated as a constant when integrating with respect to $x$, and vice versa.

STEP 2

1. Evaluate the integral with respect to $x$.
2. Evaluate the integral with respect to $y$.
3. Combine the results.

STEP 3

Evaluate the integral with respect to $x$:
$$\int e^{3x}(3yx^2 + 2xy + y^3) \, dx$$ To solve this, we integrate term by term:
- For $\int e^{3x} \cdot 3yx^2 \, dx$, use integration by parts.
- For $\int e^{3x} \cdot 2xy \, dx$, use integration by parts.
- For $\int e^{3x} \cdot y^3 \, dx$, recognize it as a simple exponential integral.

STEP 4

Perform integration by parts for the first term $\int e^{3x} \cdot 3yx^2 \, dx$:
Let $u = 3yx^2$ and $dv = e^{3x} \, dx$.
Then, $du = 6yx \, dx$ and $v = \frac{1}{3}e^{3x}$.
Apply integration by parts:
$$\int u \, dv = uv - \int v \, du$$ $$= \left(3yx^2 \cdot \frac{1}{3}e^{3x}\right) - \int \left(\frac{1}{3}e^{3x} \cdot 6yx\right) \, dx$$ $$= yx^2e^{3x} - 2y \int xe^{3x} \, dx$$

STEP 5

Perform integration by parts for the remaining integral $\int xe^{3x} \, dx$ from the previous step:
Let $u = x$ and $dv = e^{3x} \, dx$.
Then, $du = dx$ and $v = \frac{1}{3}e^{3x}$.
Apply integration by parts:
$$\int u \, dv = uv - \int v \, du$$ $$= \left(x \cdot \frac{1}{3}e^{3x}\right) - \int \frac{1}{3}e^{3x} \, dx$$ $$= \frac{1}{3}xe^{3x} - \frac{1}{9}e^{3x}$$ Substitute back:
$$yx^2e^{3x} - 2y\left(\frac{1}{3}xe^{3x} - \frac{1}{9}e^{3x}\right)$$ $$= yx^2e^{3x} - \frac{2}{3}yx e^{3x} + \frac{2}{9}ye^{3x}$$

STEP 6

Evaluate the integral $\int e^{3x} \cdot 2xy \, dx$:
This is similar to the previous integration by parts:
$$\int e^{3x} \cdot 2xy \, dx = 2y \left(\frac{1}{3}xe^{3x} - \frac{1}{9}e^{3x}\right)$$ $$= \frac{2}{3}yxe^{3x} - \frac{2}{9}ye^{3x}$$

STEP 7

Evaluate the integral $\int e^{3x} \cdot y^3 \, dx$:
This is a simple exponential integral:
$$\int e^{3x} \cdot y^3 \, dx = \frac{1}{3}y^3e^{3x}$$

STEP 8

Combine the results of the integrals with respect to $x$:
$$yx^2e^{3x} - \frac{2}{3}yx e^{3x} + \frac{2}{9}ye^{3x} + \frac{2}{3}yxe^{3x} - \frac{2}{9}ye^{3x} + \frac{1}{3}y^3e^{3x}$$ $$= yx^2e^{3x} + \frac{1}{3}y^3e^{3x}$$

STEP 9

Evaluate the integral with respect to $y$:
$$\int e^{3x}(x^2 + y^2) \, dy$$ This can be split into two separate integrals:
$$\int e^{3x}x^2 \, dy + \int e^{3x}y^2 \, dy$$

STEP 10

Evaluate $\int e^{3x}x^2 \, dy$:
Since $e^{3x}x^2$ is independent of $y$, it can be treated as a constant:
$$e^{3x}x^2 \int \, dy = e^{3x}x^2 y$$

STEP 11

Evaluate $\int e^{3x}y^2 \, dy$:
$$\int e^{3x}y^2 \, dy = e^{3x} \cdot \frac{y^3}{3}$$

STEP 12

Combine the results of the integrals with respect to $y$:
$$e^{3x}x^2 y + e^{3x} \cdot \frac{y^3}{3}$$

Combine the results from both integrals:
The final result is:
$$yx^2e^{3x} + \frac{1}{3}y^3e^{3x} + e^{3x}x^2 y + e^{3x} \cdot \frac{y^3}{3}$$ Simplify if possible:
$$2yx^2e^{3x} + \frac{2}{3}y^3e^{3x}$$ The solution to the problem is:
$$2yx^2e^{3x} + \frac{2}{3}y^3e^{3x}$$