Math

QuestionCalculate the integral 054x28xdx\int_{0}^{5} 4 x^{2} 8 x \, dx.

Studdy Solution

STEP 1

Assumptions1. We are given a definite integral with limits from0 to5. . The integrand is a polynomial function, specifically 4x8x4x^ \cdot8x.

STEP 2

First, simplify the integrand. The integrand 4x28x4x^2 \cdot8x can be simplified by multiplying the coefficients and adding the exponents of xx.
4x^2 \cdot8x =32x^

STEP 3

Now, we need to find the antiderivative of the simplified integrand. The antiderivative of xnx^n is 1n+1xn+1\frac{1}{n+1}x^{n+1}.
\int32x^3 \, dx =32 \int x^3 \, dx =32 \cdot \frac{1}{}x^ =8x^

STEP 4

Next, apply the Fundamental Theorem of Calculus, which states that the definite integral of a function from a to b is equal to the antiderivative evaluated at b minus the antiderivative evaluated at a.
032x3dx=8x40\int_{0}^{}32x^3 \, dx =8x^4 \Bigg|_{0}^{}

STEP 5

Evaluate the antiderivative at the upper limit and at the lower limit.
8x405=8548048x^4 \Bigg|_{0}^{5} =8 \cdot5^4 -8 \cdot0^4

STEP 6

Calculate the definite integral.
854804=86250=50008 \cdot5^4 -8 \cdot0^4 =8 \cdot625 -0 =5000The value of the definite integral 054x28x\int_{0}^{5}4 x^{2}8 x is5000.

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