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Solve the definite integral.

-112xcosxdx

Short Answer

Expert verified

The solution isln(2)ln(2)+12ln(2)cos1+sin(1)ln(2)-12ln(2)cos(-1)+sin(-1)ln(2).

Step by step solution

01

Step 1. Given information.

The given integral is-112xcosxdx.

02

Step 2. First, solve the indefinite integral.

I=2xcosxdx=cosx2xdx-ddx(cosx)2xdxdx=cosx·2xln(2)--sinx·2xln(2)dx=cosx·2xln(2)+sinxln(2)2xdx-ddx(sinx)2xdxdx=cosx·2xln(2)+sinxln(2)·2xln(2)-cosx·2xln(2)dx=cosx·2xln(2)+sinxln(2)·2xln(2)-1ln22xcosxdx

03

Step 3. Continued the above solution.

I=cosx·2xln(2)+sinxln(2)·2xln(2)-1ln22xcosxdxI=cosx·2xln(2)+sinxln(2)·2xln(2)-1ln2II+1ln(2)I=cosx·2xln(2)+sinxln(2)·2xln(2)I1+1ln(2)=cosx·2xln(2)+sinxln(2)·2xln(2)Iln(2)+1ln(2)=cosx·2xln(2)+sinxln(2)·2xln(2)I=ln(2)ln(2)+1cosx·2xln(2)+sinxln(2)·2xln(2)

04

Step 4. Now apply the limit of integration.

-112xcosxdx=ln(2)ln(2)+1cosx·2xln(2)+sinxln(2)·2xln(2)-11=ln(2)ln(2)+1cos(1)·21ln(2)+sin(1)ln(2)·21ln(2)-cos(-1)·2-1ln(2)+sin(-1)ln(2)·2(-1)ln(2)=ln(2)ln(2)+12ln(2)cos1+sin(1)ln(2)-cos(-1)·12ln(2)+sin(-1)ln(2)·12ln(2)=ln(2)ln(2)+12ln(2)cos1+sin(1)ln(2)-12ln(2)cos(-1)+sin(-1)ln(2)

05

Step 5. Simplified answer.

Hence, the required value is ln(2)ln(2)+12ln(2)cos1+sin(1)ln(2)-12ln(2)cos(-1)+sin(-1)ln(2).

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