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integrate int0^(2pi) e^x . sin (pi/4 + x...

integrate `int_0^(2pi) e^x . sin (pi/4 + x/2) dx`

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Given
`int_0^(2pi) e^x . sin (pi/4 + x/2) dx`
`I=int_0^(2pi) e^x . sin (pi/4 + x/2) dx`
`I=sin (pi/4 + x/2)e^x-(1/2)int_0^(2pi) cos (pi/4 + x/2) e^x dx`
`I=sin (pi/4 + x/2)e^x-(1/2)[cos (pi/4 + x/2) e^x +(1/2)int_0^(2pi) e^x .sin (pi/4 + x/2) dx]`
`I=sin (pi/4 + x/2)e^x-(1/2)cos (pi/4 + x/2) e^x -(1/4)int_0^(2pi) e^x .sin (pi/4 + x/2) dx]`
`I=sin (pi/4 + x/2)e^x-(1/2)cos (pi/4 + x/2) e^x -(1/4)I`
Hence after putting lower and upper limits we get
...
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RD SHARMA-DEFINITE INTEGRALS-Solved Examples And Exercises
  1. Evaluate: int0^(pi//2)1/(3+2cosx)dx

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  2. Evaluate: int(pi//4)^(pi//2)cos2xlogsinxdx

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  3. integrate int0^(2pi) e^x . sin (pi/4 + x/2) dx

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  4. Evaluate: int0^1(2x)/(5x^2+1)dx

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  5. Evaluate:(i) int0^(pi//6)(2+3x^2)dx

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  6. Evaluate: (i)int0^4 1/(x+sqrt(x))dx (ii) int0^1(2x)/(5x^2+1)dx

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  7. Evaluate the following definite integrals (1-58): int1^2 1/(sqrt((x...

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  8. If (x) is of the form f(x)=a+b x+c x^2, show that int0^1f(x)d...

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  9. Evaluate: int0^oo1/((x^2+a^2)(x^2+b^2))dxdot

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  10. Evaluate the following integrals: int0^(pi//6)cos^(-3)2thetasin2th...

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  11. Evaluate the following integrals: int0^((pi)^(2//3))sqrt(x)cos^2x^(...

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  12. Evaluate the following integrals: int0^1(24x^3)/((1+x^2)^4)dx

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  13. Evaluate the following integrals: int0^1sqrt((1-x)/(1+x))dx

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  14. If In=int0^(pi//4)tan^("n")x dx , prove that In+I(n-2)=1/(n-1)dot

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  15. If In=int0^(pi//4)tan^n x dx , show that 1/(I2+I4),1/(I3+I5),1/(I4+...

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  16. Evaluate: int0^(pi//2)1/((a^2cos^2x+b^2sin^2x)^2)dx

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  17. Evaluate: int0^1x(tan^(-1)x)^2dx

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  18. Evaluate: int0^a1/((x^2+a^2))dx (ii) int0^oo(x^2)/((a^2+x^2)^(5//2))dx

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  19. Evaluate: int0^1xsqrt((1-x^2)/(1+x^2))dx

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  20. Evaluate the following integrals: int0^a(sin)^(-1)sqrt(x/(a+x))dx

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