Evaluate the Integral

Problem

(∫_^)(8*x3−9*x2+4*d(x))

Solution

1. Apply the sum rule for integration, which allows the integral of a sum to be written as the sum of the integrals of each individual term.

(∫_^)(8*x3*d(x))−(∫_^)(9*x2*d(x))+(∫_^)(4*d(x))

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

(8*x(3+1))/(3+1)−(9*x(2+1))/(2+1)+4*x+C

3. Simplify the coefficients by performing the division.

(8*x4)/4−(9*x3)/3+4*x+C

4. Reduce the fractions to their simplest form.

2*x4−3*x3+4*x+C

Final Answer

(∫_^)(8*x3−9*x2+4*d(x))=2*x4−3*x3+4*x+C

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