use product-to-sum
formula sin u cos v =
1/2 [sin(u+v)
+ sin(u-v)]
so you get
1/2 int: (sin 15x) - (sin5x) dx
split it
1/2 int: sin 15x dx
- 1/2 int: sin 5x dx
using substitution you can conclude that
1/30 int: sin u du
- 1/10 in sin w dw
(you get the fraction change when you set dx=
du
and dw)
so then
- (cos u)/30 + (cos w)/10
replace the substitution
(cos 5x)/10 - (cos 15x)/30 + Constant
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∫ 10x dx Factor out the constant: 10 ∫ x dx Therefore, by the power rule, we obtain: 10x(1 + 1)/(1 + 1) + k = 10x²/2 + k = 5x² + k
5x-15x+10 = 10-10x when simplified Assuming a request to solve the equation: 5x -15x + 10 = 0 5x - 15x + 10 = 0 5x - 15x = -10 or (multiplying everything by -1 gives you -5x + 15x = 10) so 15x - 5x = 10 10x = 10 x =1
integration by parts. Let u=lnx, dv=xdx-->du=(1/x)dx, v=.5x^2. Integral of (xlnxdx)=lnx*.5x^2-integral of .5x^2(1/x)dx=lnx*.5x^2-integral of .5xdx=lnx*.5x^2-(1/6)x^3. That's it.
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