integrate:cos^10xdx
even with the previous hint a tutor here gave me i still don,t know it
hmmm lets see
using demovire theorm
z+1/z=2cosx and z^n+1/z^n=2cosnx
in our case
2cosx=(z+1/z)
pascal triangle
(z+1/z)^10
from left to right are
1,10,45,120,210,252,210,120,45,10 and 1
the coeffient now 1st,2nd---last
1st term:z^10
2ndterm:10z^9(1/z)=10z^8
3rdterm:45z^8(1/z)^2=45z^6
4thterm:120z^7(1/z)^3=120z^4
5thterm:210z^6(1/z)^2=210z^2
6thterm:252z^5(1/z)^5=252
7thterm:210z^4(1/z)^6=210/z^2
8thterm:120z^3(1/7)^7=120/z^4
9thterm:45z^2(1/z)^8=45/z^6
10thterm:10z(1/z)^9=10/z^8
11term:1/z^10
then
z^10+10z^8+45z^6+120z^4+210z^2+252+120/z^2+120/z^4+45/z^6+10/z^8+1/z^10
arrange them
z^10+1/z^10+10z^8+10/z^8+45z^6+45/z^6+120/z^4+120/z^4+210z^2+210/z^2+252
factor out
(z^10+1/z^10)+10(z^8+1/z^8)+45(z^6+1/z^6)+120(z^4+1/z^4)+210(z^2+1/z^2)+252
recap
z^n+1/z^n=2cosx
2cos10x+20cos8x+90cos6x+240cos4x+420cos2x+252
1024cos^10x=2cos10x+20cos8x+90cos6x+240cos4x+420cos2x+252
now divide by 1024
1/512cos10x+5/256cos8x+45/512cos8x+15/64cos4x+105/256cos2x+63/256
now just integrate them term by term
i need a big thanks
thank you very much sir collins
you are welcome read out calculus made easy on schaum's outline these book was recommende to me by my favorite tutor here STEVE...that book is a life saving one if you can not get one locally use amazon..steve favorite statement
google is your friend
try it out
To integrate cos^10(x), you can use the following steps:
Step 1: Use the power reduction formula to rewrite cos^10(x) as a product of cos^2(x).
cos^10(x) = (cos^2(x))^5
Step 2: Apply the power rule for integration to solve the integral of (cos^2(x))^5.
∫(cos^2(x))^5 dx
Step 3: Rewrite (cos^2(x))^5 as (1 - sin^2(x))^5.
∫(1 - sin^2(x))^5 dx
Step 4: Expand the expression using the binomial theorem.
∫(1 - 5sin^2(x) + 10sin^4(x) - 10sin^6(x) + 5sin^8(x) - sin^10(x)) dx
Step 5: Integrate each term separately using the power rule for integration.
∫1 dx - ∫5sin^2(x) dx + ∫10sin^4(x) dx - ∫10sin^6(x) dx + ∫5sin^8(x) dx - ∫sin^10(x) dx
Step 6: Evaluate each integral.
x - (5/3)sin^3(x) + (10/5)sin^5(x) - (10/7)sin^7(x) + (5/9)sin^9(x) - (1/11)sin^11(x) + C
So, the integral of cos^10(x) is:
x - (5/3)sin^3(x) + (2/5)sin^5(x) - (10/7)sin^7(x) + (5/9)sin^9(x) - (1/11)sin^11(x) + C, where C is the constant of integration.