Evaluate the Integral integral of sec(x)^6 with respect to x

Problem

(∫_^)(sec6(x)*d(x))

Solution

1. Identify the strategy for an even power of secant by splitting off a factor of sec2(x) to serve as the derivative of tan(x)

(∫_^)(sec4(x)*sec2(x)*d(x))

2. Apply the identity sec2(x)=1+tan2(x) to rewrite the remaining secant terms in terms of tangent.

(∫_^)((1+tan2(x))2*sec2(x)*d(x))

3. Expand the squared binomial inside the integrand.

(∫_^)((1+2*tan2(x)+tan4(x))*sec2(x)*d(x))

4. Substitute u=tan(x) which implies d(u)=sec2(x)*d(x)

(∫_^)((1+2*u2+u4)*d(u))

5. Integrate the polynomial with respect to u using the power rule.

u+(2*u3)/3+(u5)/5+C

6. Back-substitute u=tan(x) to return to the original variable.

tan(x)+(2*tan3(x))/3+tan5(x)/5+C

Final Answer

(∫_^)(sec6(x)*d(x))=tan(x)+2/3*tan3(x)+1/5*tan5(x)+C

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