Evaluate the Integral

Problem

(∫_^)((7*x2−5)*(3*x3+4)*d(x))

Solution

1. Expand the integrand by multiplying the two binomials using the FOIL method.

(7*x2)*(3*x3)+(7*x2)*(4)+(−5)*(3*x3)+(−5)*(4)

2. Simplify the resulting expression to prepare for integration.

21*x5+28*x2−15*x3−20

3. Rearrange the terms in descending order of their exponents.

21*x5−15*x3+28*x2−20

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

(21*x6)/6−(15*x4)/4+(28*x3)/3−20*x+C

5. Simplify the coefficients by reducing the fractions where possible.

(7*x6)/2−(15*x4)/4+(28*x3)/3−20*x+C

Final Answer

(∫_^)((7*x2−5)*(3*x3+4)*d(x))=(7*x6)/2−(15*x4)/4+(28*x3)/3−20*x+C

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