Evaluate the Integral

Problem

(∫_0^2)((1−4*x)^2*d(x))

Solution

1. Expand the integrand by squaring the binomial (1−4*x)^2 using the formula (a−b)^2=a^2−2*a*b+b^2

(1−4*x)^2=1−8*x+16*x^2

2. Rewrite the definite integral with the expanded polynomial.

(∫_0^2)((1−8*x+16*x^2)*d(x))

3. Find the antiderivative of each term using the power rule (∫_^)(x^n*d(x))=(x^(n+1))/(n+1)

(∫_^)((1−8*x+16*x^2)*d(x))=x−(8*x^2)/2+(16*x^3)/3

4. Simplify the coefficients of the antiderivative.

x−4*x^2+(16*x^3)/3

5. Apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper limit 2 and the lower limit 0

(2−4*(2)^2+(16*(2)^3)/3)−(0−4*(0)^2+(16*(0)^3)/3)

6. Calculate the numerical values.

(2−16+128/3)−0

7. Combine the terms into a single fraction.

−14+128/3

(−42)/3+128/3

86/3

Final Answer

(∫_0^2)((1−4*x)^2*d(x))=86/3

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