Evaluate the Integral

Problem

(∫_^)(sec2(x)*tan3(x)*d(x))

Solution

1. Identify the substitution. Since the derivative of tan(x) is sec2(x) let u=tan(x)

2. Calculate the differential d(u)

d(u)=sec2(x)*d(x)

3. Substitute u and d(u) into the integral.

(∫_^)(u3*d(u))

4. Apply the power rule for integration, which states (∫_^)(un*d(u))=(u(n+1))/(n+1)+C

(u4)/4+C

5. Back-substitute the original variable x using u=tan(x)

tan4(x)/4+C

Final Answer

(∫_^)(sec2(x)*tan3(x)*d(x))=tan4(x)/4+C

Want more problems? Check here!