Recall that the integral Integral (5 cos^31 - 6 cos^4 t) dt is a pain to compute
ID: 1720469 • Letter: R
Question
Recall that the integral Integral (5 cos^31 - 6 cos^4 t) dt is a pain to compute. So instead of computing it as is, we will change the integrand 5 cos^3 t - 6 cos^4 t to be a linear combination of vectors in C, which will be easier to integrate. To do this, consider the matrix P, with columns the B coordinates of vectors in C. That is, Use this to find P-1 (no need to do this by hand - use technology), and then use P-1 to change 5cos^3 t - 6 cos^4 t into a linear combination of vectors in B. First write 5cos^3 t - 6 cos^4 t as a vector v R^7 with the basis B, and then find P^-1 v. The vector P^-1v will also be in M^7, but with the basis C. Now write the integrand as linear combination of vectors in C, and integrate!Explanation / Answer
once you have P^-1
we calculate P^-1(v)
v here is [0,0,0,0,0,5,-12]
P^-1(v) is a vector in R^7 say
[a,b,c,d,e,f,g]
then required integral is the integral of
a +b cost + ....... g cos6t
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