Evaluate the Integral

Problem

(∫_0^1)(4*x5−5*x4*d(x))

Solution

1. Identify the integral as a definite integral of a polynomial function.

2. Apply the power rule for integration, which states that (∫_^)(xn*d(x))=(x(n+1))/(n+1) to each term of the integrand.

3. Find the antiderivative of the expression.

(∫_^)(4*x5−5*x4*d(x))=(4*x6)/6−(5*x5)/5

4. Simplify the coefficients of the antiderivative.

(2*x6)/3−x5

5. Apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper limit 1 and the lower limit 0

((2*(1)6)/3−(1)5)−((2*(0)6)/3−(0)5)

6. Calculate the numerical values.

(2/3−1)−(0)

7. Subtract the fractions to find the final result.

2/3−3/3=−1/3

Final Answer

(∫_0^1)(4*x5−5*x4*d(x))=−1/3

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