- #1

Physics Slayer

- 26

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- Homework Statement
- Need help finding the following indefinite integrals

- Relevant Equations
- -

Been struggling with a few integrals, I might post a few more once I progress further in my assignment.

$$1. \int \sqrt{tanx} + \sqrt{cotx} (dx)$$

for integral 1, I try to apply integration by parts on both ##\sqrt{tanx}## and ##\sqrt{cotx}## separately, I then get

$$\int \underbrace{\sqrt{tanx}}_{\textrm{u}}\underbrace{(dx)}_{\textrm{dv}} + \int \underbrace{\sqrt{cotx}}_{\textrm{u}}\underbrace{(dx)}_{\textrm{dv}}$$

this gives,

$$=\left( x\sqrt{tanx}-\int \underbrace{x}_{\textrm{u}}.\underbrace{\frac{sec^2x}{2\sqrt{tanx}}}_{\textrm{dv}}(dx)\right) + \left(x\sqrt{cotx}-\int \underbrace{x}_{\textrm{u}}.\underbrace{\frac{(-coesc^2x)}{2\sqrt{cotx}}}_{\textrm{dv}}(dx)\right)$$

after applying integration by parts again I sadly get, RHS = LHS

$$=\left(xtanx - \left(x\sqrt{tanx}-\int \sqrt{tanx}(dx)\right)\right) + \left(x\sqrt{cotx}-\left(x\sqrt{cotx}-\int\sqrt{cotx}(dx)\right)\right)$$

$$=\int\sqrt{tanx}(dx)+\int\sqrt{cotx}(dx)$$

this basically lead me back to the beginning:(

I also tried using a the substitution, ##t^2 = tan(x)## but even this brought me back to where I started,

$$\int \sqrt{tanx}+\frac{1}{\sqrt{tanx}} (dx) = \int\frac{tanx+1}{\sqrt{tanx}}(dx)$$

using the above sub,

$$\int\frac{t^2+1}{t}\frac{2t}{1+t^4}(dt) = 2\int\frac{t^2+1}{t^4+1}dt$$

now ##u=t^2##

$$=\int\frac{u+1}{u^2+1}\frac{du}{\sqrt{u}}=\int\frac{\sqrt{u}}{u^2+1}+\frac{1}{\sqrt{u}(u^2+1)}(du)$$

and finally if you use the trig sub, ##u = tan\theta## you end up with,

$$=\int \sqrt{tan\theta}+\sqrt{cot\theta}(d\theta)$$

I usually don't like asking for help while solving integrals, there is a different satisfaction one gets when they finally get the "aha" moment and then solve the problem, but these are literally getting in the way of my life, I don't want entire solutions, a hint or a reassurance that I am thinking in the right direction will do :)

I need help in these two as well, but I'll show my working for them once I get this pesky one out of the way.

$$2. \int \frac{1}{x^{1/2} + x^{1/3}} (dx)$$

$$3. \int \frac{cos(2x) - sin(2\phi)}{cos(x) - sin(\phi)} (dx)$$

$$1. \int \sqrt{tanx} + \sqrt{cotx} (dx)$$

**Attempt1**:for integral 1, I try to apply integration by parts on both ##\sqrt{tanx}## and ##\sqrt{cotx}## separately, I then get

$$\int \underbrace{\sqrt{tanx}}_{\textrm{u}}\underbrace{(dx)}_{\textrm{dv}} + \int \underbrace{\sqrt{cotx}}_{\textrm{u}}\underbrace{(dx)}_{\textrm{dv}}$$

this gives,

$$=\left( x\sqrt{tanx}-\int \underbrace{x}_{\textrm{u}}.\underbrace{\frac{sec^2x}{2\sqrt{tanx}}}_{\textrm{dv}}(dx)\right) + \left(x\sqrt{cotx}-\int \underbrace{x}_{\textrm{u}}.\underbrace{\frac{(-coesc^2x)}{2\sqrt{cotx}}}_{\textrm{dv}}(dx)\right)$$

after applying integration by parts again I sadly get, RHS = LHS

$$=\left(xtanx - \left(x\sqrt{tanx}-\int \sqrt{tanx}(dx)\right)\right) + \left(x\sqrt{cotx}-\left(x\sqrt{cotx}-\int\sqrt{cotx}(dx)\right)\right)$$

$$=\int\sqrt{tanx}(dx)+\int\sqrt{cotx}(dx)$$

this basically lead me back to the beginning:(

**Attempt2**:I also tried using a the substitution, ##t^2 = tan(x)## but even this brought me back to where I started,

$$\int \sqrt{tanx}+\frac{1}{\sqrt{tanx}} (dx) = \int\frac{tanx+1}{\sqrt{tanx}}(dx)$$

using the above sub,

$$\int\frac{t^2+1}{t}\frac{2t}{1+t^4}(dt) = 2\int\frac{t^2+1}{t^4+1}dt$$

now ##u=t^2##

$$=\int\frac{u+1}{u^2+1}\frac{du}{\sqrt{u}}=\int\frac{\sqrt{u}}{u^2+1}+\frac{1}{\sqrt{u}(u^2+1)}(du)$$

and finally if you use the trig sub, ##u = tan\theta## you end up with,

$$=\int \sqrt{tan\theta}+\sqrt{cot\theta}(d\theta)$$

I usually don't like asking for help while solving integrals, there is a different satisfaction one gets when they finally get the "aha" moment and then solve the problem, but these are literally getting in the way of my life, I don't want entire solutions, a hint or a reassurance that I am thinking in the right direction will do :)

I need help in these two as well, but I'll show my working for them once I get this pesky one out of the way.

$$2. \int \frac{1}{x^{1/2} + x^{1/3}} (dx)$$

$$3. \int \frac{cos(2x) - sin(2\phi)}{cos(x) - sin(\phi)} (dx)$$

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