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How do you evaluate the definite integral #int cos2theta# from #[0,pi/4]#?

Answer 1

Isabella Ackerman

#1/2#

#int_0^(pi/4)cos2thetad theta#

now

#d/(d theta)(sin2 theta)=2cos2 theta#

#int_0^(pi/4)cos2thetad theta=1/2[sin2 theta]_0^(pi/4)#

#1/2[sin2 theta]_0^(pi/4)=1/2[sin 2(pi/4)-cancel(sin (2xx0))]#

#=1/2sin(pi/2)=1/2xx1=1/2#

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Answer 2

Emma Aldridge

To evaluate the definite integral ∫ cos^2(θ) from 0 to π/4, you can use the trigonometric identity cos^2(θ) = (1 + cos(2θ))/2. Then integrate (1 + cos(2θ))/2 with respect to θ from 0 to π/4. After integrating, substitute the upper limit of integration (π/4) and subtract the result when substituting the lower limit of integration (0). This will give you the value of the definite integral.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.