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What is the integral using the substitution u=sin(2x)?

find #int_0^(pi/4)sin^4(2x)cos(2x)dx# using the substitution #u=sin(2x)#

Answer 1

#I=1/10#

We want to integrate

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

Make a substitution #color(blue)(u=sin(2x)=>du=2cos(2x)dx# New limits #color(blue)(x=0=>u=0# and #color(blue)(x=pi/4=>u=1#

#I=1/2int_0^1u^4du=[1/10u^5]_0^1=1/10#

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Answer 2

Isabella Ackerman

The integral using the substitution ( u = \sin(2x) ) involves trigonometric functions and will depend on the limits of integration and the function being integrated. Without specifying the function and the limits of integration, the integral cannot be determined.

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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.