How do you evaluate the integral? #int_0^(pi/2)cos^(3)x sin2x dx#

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Answer 1 · 2017-10-03T22:24:12

Substitute #2sin(x)cos(x)# for #sin(2x)#
Perform a "u" substitution and change of limits
Integrate
Evaluate at the limits:

Given #int_0^(pi/2)cos^3(x) sin(2x) dx#

Substitute #2sin(x)cos(x)# for #sin(2x)#

#2int_0^(pi/2)cos^4(x) sin(x) dx#

Let #u = cos(x)#, then #(du)/dx = -sin(x)#

A more suitable form for substitution:

#sin(x)dx = -du#

#b = cos(pi/2) = 0#

#a = cos(0) = 1#

#-2int_1^0u^4(x)du#

Negate and flip the limits:

#2int_0^1u^4(x)du = 2/5(u^5]_0^1 = 2/5(1^5-0^5)#

#int_0^(pi/2)cos^3(x) sin(2x) dx = 2/5#