integrate. (1):sin2xcos3x
(2):5(cos3x)^2
To integrate the given functions, we need to use integral rules and techniques. Let's solve each integration step by step.
(1) ∫sin^2(x)cos^3(x) dx:
We can use a trigonometric identity to simplify this expression:
sin^2(x) = (1 - cos(2x))/2
Now, we substitute the simplified expression into the integral:
∫ [(1 - cos(2x))/2]cos^3(x) dx
Next, we distribute the factor of 1/2 inside the integral:
(1/2)∫[cos^3(x) - cos^3(2x)/2] dx
Now, we can divide this integral into two separate integrals:
(1/2)∫[cos^3(x)] dx - (1/4)∫[cos^3(2x)] dx
To integrate cos^3(x), we can use a trigonometric identity:
cos^3(x) = (3cos(x) + cos(3x))/4
So, let's substitute this identity into the first integral:
(1/2)∫[(3cos(x) + cos(3x))/4] dx
Integrating each term separately, we get:
(1/2)(3/4) ∫[cos(x)] dx + (1/2)(1/4) ∫[cos(3x)] dx
Simplifying further, we have:
(3/8) ∫[cos(x)] dx + (1/8) ∫[cos(3x)] dx
Integrating cos(x) and cos(3x) individually:
(3/8)sin(x) + (1/8)(1/3)sin(3x)
Simplifying the expressions:
(3/8)sin(x) + (1/24)sin(3x)
Therefore, the final answer to the integral of sin^2(x)cos^3(x) is:
(3/8)sin(x) + (1/24)sin(3x) + C, where C is the constant of integration.
(2) ∫5(cos^3(3x)) dx:
We can use the same trigonometric identity we used in the previous problem:
cos^3(x) = (3cos(x) + cos(3x))/4
Now, substitute the simplified expression into the integral:
∫5[(3cos(3x) + cos(9x))/4] dx
Next, we distribute the factor of 5/4 inside the integral:
(5/4)∫[(3cos(3x) + cos(9x))] dx
Integrating each term separately, we get:
(5/4)(3/3) ∫[cos(3x)] dx + (5/4)(1/9) ∫[cos(9x)] dx
Simplifying further:
(15/12) ∫[cos(3x)] dx + (5/36) ∫[cos(9x)] dx
Integrating cos(3x) and cos(9x) individually:
(15/12)(1/3)sin(3x) + (5/36)(1/9)sin(9x)
Simplifying the expressions:
(5/8)sin(3x) + (5/324)sin(9x)
Therefore, the final answer to the integral of 5(cos^3(3x))^2 is:
(5/8)sin(3x) + (5/324)sin(9x) + C, where C is the constant of integration.
hmmm, graph that first function.
sin a cos b = (1/2)sin(a+b)+(1/2) sin(a-b)
(remember about beat frequencies :)
so
(1/2)[ sin(5x) dx - sin(x) dx ]
ok?
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let y = 3x
then dy = 3 dx or dx = (1/3) dy
so
(5/3) [ cos^2 y dy ]
5(cos3x)^2
but you need a siny to form the full derivative
cos^2 u = (1+cos 2u)/2
so, we have
5/2 (cos6x+1)
Now you have
5/2 (1/6 sin6x + x) + C