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Expert: - 4/30/2008


Question
how to intergrate

1) sin(14x) cos(5x) dx

2) 2.7^2x+7 dx  

Answer
1) Use three trig identities and simplify the first one.  
The identies are sin^2(x) + cos^2(x) = 1,
sin(a+b) = sin(a)cos(b) + sin(b)cos(a), and
cos(a+b) = cos(a)cos(b) - sin(a)sin(b).

I'm not entirely sure how the problem will turn out, but it will take a few iterations to do.  I believe you have to simplify it all down to sin(x) and cos(x) to various powers.  Once you have done this, you need to use the formula sin^2(x) + cos^2(x) = 1 to solve it for
cos^2(x) or sin^2 and substitute it back in.

You should do this with the one that has the odd power.  

Once the power on that trig function is one, a u-substitution can be done with u being the other trig function.  Note that this will give you several integrals that each requires a u-substitution.

2) The integral(e^x)dx is e^x + C, so the integral(a^x)dx  is
(a^x)ln(a) + C.

Calculus

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