Use a table of integrals to evaluate the definite integral. \int_{0}^{\frac{\pi}{2}}x\sin

Stefan Hendricks

Stefan Hendricks

Answered question

2021-12-29

Use a table of integrals to evaluate the definite integral.
0π2xsin2xdx

Answer & Explanation

Philip Williams

Philip Williams

Beginner2021-12-30Added 39 answers

Step 1
Given: I=0π2xsin(2x)dx
for evaluating given integral, we use integral by parts theorem
according to this theorem
(f(x))g(x)dx=g(x)f(x)dx[(g(x))f(x)dx]dx+c
Step 2
so,
0π2xsin(2x)=[xsin2xdx[1sin2xdx]dx]0π2
(sinkxdx=coskxk+c)
=[x(cos2x2)(cos2x2)dx]0π2
(coskxdx=sinkxk+c)
=[xcos2x2+12sin2x2]0π2
=[2xcos2x+sin2x4]0π2
=14[(2(π2)cos(π)+sin(π))(2(0)cos(0)+sin(0))]
=14[π(1)+0+00]
=14(π)
=π4
hence, given integral is equal to
veiga34

veiga34

Beginner2021-12-31Added 32 answers

xsin(2x)dx
=xcos(2x)2cos(2x)2dx
cos(2x)2dx
=14cos(u)du
cos(u)du
=sin(u)
14cos(u)du
=sin(u)4
=sin(2x)4
xcos(2x)2cos(2x)2dx
=sin(2x)4xcos(2x)2
xsin(2x)dx
=sin(2x)4xcos(2x)2+C
=sin(2x)2xcos(2x)4+C
karton

karton

Expert2022-01-04Added 613 answers

0π2x×sin(2x)dx
Evaluate the indefinite integral
x×sin(2x)dxUse the partial integrationx×(cos(2x)2)cos(2x)2dxx×(cos(2x)2)1×(12)×cos(2x)dxx×(cos(2x)2)+12×cos(t)2dtx×(cos(2x)2)+12×12×cos(t)dtx×(cos(2x)2)+14×sin(t)x×(cos(2x)2)+14×sin(2x)x×cos(2x)2+sin(2x)4(x×cos(2x)2+sin(2x)4)|0π2π2×cos(2×π2)2+sin(2×π2)4(0cos(2×0)2+sin(2×0)4)
Simplify
Solution
π4

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