Evaluate the Integral integral of sec(x)^5 with respect to x

Problem

(∫_^)(sec5(x)*d(x))

Solution

1. Apply the reduction formula for powers of secant, which is (∫_^)(secn(x)*d(x))=(sec(x)(n−2)*tan(x))/(n−1)+(n−2)/(n−1)*(∫_^)(sec(x)(n−2)*d(x))

2. Substitute n=5 into the reduction formula to reduce the power from 5 to 3.

(∫_^)(sec5(x)*d(x))=(sec3(x)*tan(x))/4+3/4*(∫_^)(sec3(x)*d(x))

3. Apply the reduction formula again to the integral (∫_^)(sec3(x)*d(x)) using n=3

(∫_^)(sec3(x)*d(x))=(sec(x)*tan(x))/2+1/2*(∫_^)(sec(x)*d(x))

4. Evaluate the basic integral (∫_^)(sec(x)*d(x))

(∫_^)(sec(x)*d(x))=ln(sec(x)+tan(x))

5. Substitute the result from step 4 back into the expression for (∫_^)(sec3(x)*d(x))

(∫_^)(sec3(x)*d(x))=(sec(x)*tan(x))/2+1/2*ln(sec(x)+tan(x))

6. Substitute the result from step 5 back into the expression for (∫_^)(sec5(x)*d(x))

(∫_^)(sec5(x)*d(x))=(sec3(x)*tan(x))/4+3/4*((sec(x)*tan(x))/2+1/2*ln(sec(x)+tan(x)))

7. Distribute the constant and add the constant of integration C

(∫_^)(sec5(x)*d(x))=1/4*sec3(x)*tan(x)+3/8*sec(x)*tan(x)+3/8*ln(sec(x)+tan(x))+C

Final Answer

(∫_^)(sec5(x)*d(x))=1/4*sec3(x)*tan(x)+3/8*sec(x)*tan(x)+3/8*ln(sec(x)+tan(x))+C

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