Evaluate the Integral

Problem

(∫_^)(3*x*(x2−3)7*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of the inner function x2−3 is a multiple of the outer factor x

2. Define the substitution variable u=x2−3

3. Differentiate u with respect to x to find d(u)=2*x*d(x) which implies x*d(x)=1/2*d(u)

4. Substitute the expressions for u and d(x) into the integral.

(∫_^)(3*(x2−3)7*x*d(x))=(∫_^)(3*u71/2*d(u))

5. Simplify the constant factors outside the integral.

3/2*(∫_^)(u7*d(u))

6. Integrate using the power rule (∫_^)(un*d(u))=(u(n+1))/(n+1)+C

3/2(u8)/8+C

7. Simplify the resulting fraction.

3/16*u8+C

8. Back-substitute the original expression x2−3 for u

3/16*(x2−3)8+C

Final Answer

(∫_^)(3*x*(x2−3)7*d(x))=3/16*(x2−3)8+C

Want more problems? Check here!