Evaluate the Integral

Problem

(∫_^)(sec2(x)*tan5(x)*d(x))

Solution

1. Identify the substitution. Since the derivative of tan(x) is sec2(x) we can use usubstitution.

2. Define the variables for substitution. Let u=tan(x)

3. Differentiate u with respect to x to find d(u)

d(u)/d(x)=sec2(x)

4. Rewrite the differential d(x) in terms of d(u)

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

5. Substitute u and d(u) into the original integral.

(∫_^)(u5*d(u))

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

(u6)/6+C

7. Back-substitute the original expression for u

tan6(x)/6+C

Final Answer

(∫_^)(sec2(x)*tan5(x)*d(x))=tan6(x)/6+C

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