# Thread: interesting simple integrals I:

1. ## interesting simple integrals I:

I will be posting some interesting integrals.Lets work them out!
1.$\displaystyle \int_0^\frac{\pi}{2} \frac{dx}{1+(\tan x)^{101}}$

2.$\displaystyle \int \cos\log x dx$

3.$\displaystyle \int_0^{\pi/4} \ln (1+\tan x)dx$

4.$\displaystyle \int_0^{100} e^{x-[x]}$

have fun solving and do post different solutions

3. ## Re: interesting simple integrals I:

Originally Posted by earthboy
I will be posting some interesting integrals.Lets work them out!
1.$\displaystyle \int_0^\frac{\pi}{2} \frac{dx}{1+(\tan x)^{101}}$

2.$\displaystyle \int \cos\log x dx$

3.$\displaystyle \int_0^{\pi/4} \ln (1+\tan x)dx$

4.$\displaystyle \int_0^{100} e^{x-[x]}$

have fun solving and do post different solutions
For the second...

\displaystyle \displaystyle \begin{align*} I &= \int{\cos{\log{x}}\,dx} \\ I &= \int{\frac{x\cos{\log{x}}}{x}\,dx} \end{align*}

Now let \displaystyle \displaystyle \begin{align*} u = \log{x} \implies du = \frac{1}{x}\,dx \end{align*} and the integral becomes

\displaystyle \displaystyle \begin{align*} I &= \int{e^u\cos{u}\,du} \\ I &= e^u\sin{u} - \int{e^u\sin{u}\,du} \\ I &= e^u\sin{u} - \left(-e^u\cos{u} - \int{-e^u\cos{u}\,du}\right) \\ I &= e^u\sin{u} + e^u\cos{u} - \int{e^u\cos{u}\,du} + c \\ I &= e^u\sin{u} + e^u\cos{u} - I + c \\ 2I &= e^u\sin{u} + e^u\cos{u} + c \\ I &= \frac{1}{2}e^u\sin{u} + \frac{1}{2}e^u\cos{u} + C \textrm{ where }C = \frac{1}{2}c \\ I &= \frac{1}{2}e^{\log{x}}\sin{\log{x}} + \frac{1}{2}e^{\log{x}}\cos{\log{x}} + C \\ I &= \frac{1}{2}x\sin{\log{x}} + \frac{1}{2}x\cos{\log{x}} + C \end{align*}

4. ## Re: interesting simple integrals I:

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