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This may look very daunting at first, but working through it slowly but carefully can provide the correct solution.
First, I will assume that there are no bounds of integration (i.e., this is an indefinite integral).
Now, the problem can be solved.
So integral of (cosx)cos(sin(x))dx can be turned into "u's", using u-substituion. Let u=sinx and du = cosxdx. We are now left with integral of cos(u)du. This is sinu. But because it is an indefinite integral the constant "C" must be added. So the final answer is integral of cosxcos(sin(x))dx=integral of cosudu=sinu+C
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Let's integrate our teams.
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