interesting simple integrals I:

**earthboy** · July 5th 2012, 03:04 AM

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

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

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

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

have fun solving and do post different solutions

**tom@ballooncalculus** · July 5th 2012, 04:13 AM

2. E.g... picture, in case it's fun

**Prove It** · July 5th 2012, 04:37 AM

Originally Posted by earthboy

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

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

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

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

have fun solving and do post different solutions

For the second...

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

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

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

**tom@ballooncalculus** · July 5th 2012, 05:51 AM

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