Math Snap

STEP 1

1. We are given the definite integral $\int_{0}^{4}\left(5 x^{4}+2\right) d x$.
2. We need to evaluate this integral over the interval from $x = 0$ to $x = 4$.

STEP 2

1. Find the indefinite integral of the function $5x^4 + 2$.
2. Evaluate the indefinite integral at the upper limit of integration.
3. Evaluate the indefinite integral at the lower limit of integration.
4. Subtract the value obtained at the lower limit from the value obtained at the upper limit to find the definite integral.

STEP 3

Find the indefinite integral of $5x^4 + 2$:
$$\int \left(5x^4 + 2\right) dx$$

STEP 4

Integrate each term separately:
$$\int 5x^4 \, dx = \frac{5}{5}x^{4+1} = x^5$$ $$\int 2 \, dx = 2x$$ Combine the results:
$$\int \left(5x^4 + 2\right) dx = x^5 + 2x + C$$

STEP 5

Evaluate the indefinite integral at the upper limit $x = 4$:
$$x^5 + 2x \bigg|_{x=4} = (4)^5 + 2(4)$$ Calculate the expression:
$$= 1024 + 8 = 1032$$

STEP 6

Evaluate the indefinite integral at the lower limit $x = 0$:
$$x^5 + 2x \bigg|_{x=0} = (0)^5 + 2(0)$$ Calculate the expression:
$$= 0 + 0 = 0$$

Subtract the value at the lower limit from the value at the upper limit:
$$1032 - 0 = 1032$$ The value of the definite integral is:
$$\boxed{1032}$$