Evaluate-0-2pi-e-x-cos-pi-4-x-2-dx-

Question Number 13623 by Tinkutara last updated on 21/May/17

Evaluate : \int_{0}^{2 π} e^{x} \cos (\frac{π}{4} + \frac{x}{2}) d x

Answered by ajfour last updated on 21/May/17

I = \int_{0}^{2 π} e^{x} \cos (\frac{π}{4} + \frac{x}{2}) d x

= [e^{2 π} (- \frac{1}{\sqrt{2}}) - \frac{1}{\sqrt{2}}] + \frac{1}{2} \int_{0}^{2 π} e^{x} \sin (\frac{π}{4} + \frac{x}{2}) d x

= - \frac{1}{\sqrt{2}} (e^{2 π} + 1) + (\frac{1}{2}) [- \frac{1}{\sqrt{2}} (e^{2 π} + 1)]

- \frac{1}{4} \int_{0}^{2 π} e^{x} \cos (\frac{π}{4} + \frac{x}{2}) d x

\Rightarrow I = - \frac{3}{2 \sqrt{2}} (e^{2 π} + 1) - \frac{I}{4}

\Rightarrow I = - \frac{3 \sqrt{2}}{5} (e^{2 π} + 1) .

Commented by ajfour last updated on 22/May/17

c o r r e c t a n s w e r p l e a s e ?

Commented by Tinkutara last updated on 22/May/17

Thanks Sir! Your answer is right .