SOLUTION: Evalute integral sin^2 (x) * cos^2 (x) dx from bounds 0 to pi/3
by using : cos^2 (x) = 1 - sin^2 (x)
Algebra.Com
Question 1191333: Evalute integral sin^2 (x) * cos^2 (x) dx from bounds 0 to pi/3
by using : cos^2 (x) = 1 - sin^2 (x)
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!
Before we get to the calculus, here are some useful power reduction trig identities.
 = \frac{1+\cos(2\text{x})}{2} = 0.5(1+\cos(2\text{x})))
 = \frac{1-\cos(2\text{x})}{2} = 0.5(1-\cos(2\text{x})))
If we were to square both sides of the second identity, then we have:
 = \frac{1-\cos(2\text{x})}{2})
\right)^2 = \left(\frac{1-\cos(2\text{x})}{2}\right)^2)
 = \frac{\left(1-\cos(2\text{x})\right)^2}{4})
 = \frac{1-2\cos(2\text{x})+\cos^2(2\text{x})}{4})
 = \frac{1-2\cos(2\text{x})+0.5(1+\cos(4\text{x}))}{4})
 = \frac{1-2\cos(2\text{x})+0.5+0.5\cos(4\text{x})}{4})
 = \frac{1.5-2\cos(2\text{x})+0.5\cos(4\text{x})}{4})
 = \frac{0.5(3-4\cos(2\text{x})+\cos(4\text{x}))}{4})
 = \frac{3-4\cos(2\text{x})+\cos(4\text{x})}{8})
which will be useful later.
-------------------------------------
Now to rewrite the integral using those trig identities and cos^2 (x) = 1 - sin^2 (x)
\cos^2(\text{x})d\text{x})
\left(1-\sin^2(\text{x})\right)d\text{x})
-\sin^4(\text{x})\right)d\text{x})
d\text{x}-\int_{0}^{\pi/3}\sin^4(\text{x})d\text{x})
}{2}d\text{x}-\int_{0}^{\pi/3}\frac{3-4\cos(2\text{x})+\cos(4\text{x})}{8}d\text{x})
I'll let you finish up from here. You'll need to use u-substitution at some parts.
RELATED QUESTIONS
Is the integral converges ??
integral [0 to pi/2] (x/cosx)sin(tgx)... (answered by Fombitz)
Sin x = 1/3 ,0 < x < pi/2
Cos y = 2/5 ,0 < x < pi/2
Find sin (x+y) (answered by ikleyn)
Evaluate the integral 1/sqr(x^2-3x+2) dx from bounds x=2.5 to... (answered by ikleyn)
cos(x+pi/2)=-sin x
(answered by math-vortex)
cos(x+pi/2) =... (answered by MathLover1)
Evaluate... (answered by Fombitz)
Evaluate {{{int(x/(1+cos^2(x)), dx, 0,... (answered by Fombitz)
Evaluate the integral of (cos^3 x)^7 dx from x=0 to... (answered by Fombitz)
Verify the expression: cos^3 + sin^2 cos = cos
Cos^3 + sin^2 cos = cos
Cos x cos^2 + (answered by Alan3354)