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Answer:
Step-by-step explanation:
So we have the integral:
First, let's use substitution to get rid of the x/2. I'm going to use the variable y. So, let y be x/2. Thus:
Therefore, the integral is:
Now, as you had done, let's expand the tangent term. However, let's do it to the fourth. Thus:
Now, we can use a variation of the trigonometric identity:
So:
Substitute this into the integral. Note that tan^4(x) is the same as (tan^2(x))^2. Thus:
Now, we can use substitution. We will use it for sec(x). Recall what the derivative of secant is. Thus:
Substitute:
Expand the binomial:
Spilt the integral:
Factor out the constant multiple:
Reverse Power Rule:
Simplify:
Distribute the 2:
Substitute back secant for u:
And substitute back 1/2x for y. Therefore:
And, finally, C:
And we're done!