Question

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Answer 1

1 - 

Identidade trigonométrica sen(2x) = 2sen(x)cos(x)

$\displaystyle \int \frac{1+\sin(2x)}{\sin^{2}(x)} \, dx \\ \\ \\ \int \frac{1}{\sin^{2}(x)} \, dx + \int \frac{\sin(2x)}{\sin^{2}(x)} \, dx \\ \\ \\ \int \frac{1}{\sin^{2}(x)} \, dx + \int \frac{2 \sin(x) \cos(x)}{\sin^{2}(x)} \, dx \\ \\ \\ - \cot(x) + \int \frac{2 \sin(x) \cos(x)}{\sin^{2}(x)} \, dx \\ \\ \\ u = \sin^{2}(x) \\ \\ du = 2\sin(x)\cos(x) \, dx \\ \\ \\ -\cot(x)+ \int \frac{du}{u} \\ \\ \\ -\cot(x)+ \int \frac{1}{u} \, du \\ \\ \\ -\cot(x)+ \ln(u)+c$

$\boxed{\boxed{-\cot(x)+\ln(\sin^{2}(x))+c}}$

2 - 

$\displaystyle \int \frac{x+1}{\sqrt{x^{2}+1}} \, dx \\ \\ \\ \int \frac{x}{\sqrt{x^{2}+1}} \, dx + \int \frac{1}{\sqrt{x^{2}+1}} \, dx \\ \\ \\ \int \frac{x}{\sqrt{x^{2}+1}} \, dx + \ln(x + \sqrt{x^{2}+1}) \\ \\ \\ u = x^{2}+1 \\ \\ du = 2x \, dx \\ \\ \\ \frac{x}{2x} \cdot \int \frac{1}{\sqrt{u}} \, du + \ln(x + \sqrt{x^{2}+1}) \\ \\ \\ \frac{1}{2} \cdot 2\sqrt{u}+\ln(x + \sqrt{x^{2}+1}) \\ \\ \\ \boxed{\boxed{\sqrt{x^{2}+1}+\ln(x + \sqrt{x^{2}+1})+c}}$

3 - 

Integral por parte

$\displaystyle \int \cos(\ln(x)) \, dx \\ \\ \\ \int \cos(\ln(x)) \cdot 1 \, dx \\ \\ \\ \cos(\ln(x)) \cdot x - \int x \cdot -\sin(\ln(x)) \cdot \frac{1}{x} \, dx \\ \\ \\ \boxed{1} \, \cos(\ln(x)) \cdot x + \int \sin(\ln(x)) \, dx \\ \\ \\ ==== \\ \\ \\ \int \sin(\ln(x)) \cdot 1 \, dx \\ \\ \\ \sin(\ln(x)) \cdot x - \int x \cos(\ln(x)) \cdot \frac{1}{x} \, dx \\ \\ \\ \boxed{2} \, \sin(\ln(x)) \cdot x - \int \cos(\ln(x)) \, dx \\ \\ \\ =====$

$\displaystyle \boxed{1} \, \, e \, \, \boxed{2} \\ \\ \\ \int \cos(\ln(x)) \, dx = \cos(\ln(x)) \cdot x + \sin(\ln(x)) \cdot x - \int \cos(\ln(x)) \, dx \\ \\ \\ \int \cos(\ln(x)) \, dx = x \cdot \cos(\ln(x)) + x \cdot \sin(\ln(x)) - \int \cos(\ln(x)) \, dx \\ \\ \\ \int \cos(\ln(x)) \, dx + \int \cos(\ln(x)) \, dx = x \cdot \cos(\ln(x)) + x \cdot \sin(\ln(x)) \\ \\ \\ 2 \cdot \int \cos(\ln(x)) \, dx = x \cdot \cos(\ln(x)) + x \cdot \sin(\ln(x))$

$\displaystyle \int \cos(\ln(x)) \, dx = \frac{x \cdot \cos(\ln(x)) + x \cdot \sin(\ln(x))}{2} \\ \\ \\ \int \cos(\ln(x)) \, dx = \frac{1}{2}x\cos(\ln(x))+\frac{1}{2}x\sin(\ln(x)) \\ \\ \\ \boxed{\boxed{\frac{1}{2}x\cos(\ln(x))+\frac{1}{2}x\sin(\ln(x))+c}}$

4 - 

$\displaystyle \int \frac{x^{4}+2x^{2}+2}{x^{2}+1} \, dx \\ \\ \\ \int \frac{x^{4}}{x^{2}+1} \, dx + \int \frac{2x^{2}}{x^{2}+1} \, dx + \int \frac{2}{x^{2}+1} \, dx \\ \\ \\ ==== \\ \\ \\ \int \frac{x^{4}}{x^{2}+1} \, dx \\ \\ \\ \int x^{2}-1+\frac{1}{x^{2}+1} \, dx \\ \\ \\ \int x^{2}-1 \, dx + \int \frac{1}{x^{2}+1} \, dx \\ \\ \\ \frac{1}{3}x^{3} - x + \arctan(x) \\ \\ \\ ====$

$\displaystyle \int \frac{2x^{2}}{x^{2}+1} \, dx \\ \\ \\ 2 \cdot \int \frac{x^{2}}{x^{2}+1} \, dx \\ \\ \\ 2 \cdot \int -\frac{1}{x^{2}+1}+1 \, dx \\ \\ \\ 2 \cdot (- \arctan(x)+x) \\ \\ \\ -2 \arctan(x)+2x \\ \\ \\ ====$

$\displaystyle \int \frac{2}{x^{2}+1} \, dx \\ \\ \\ 2 \arctan(x) \\ \\ \\ ====$

$\displaystyle \frac{1}{3}x^{3} - x + \arctan(x)-2 \arctan(x)+2x+2 \arctan(x) \\ \\ \\ \boxed{\boxed{\frac{1}{3}x^{3}+x+ \arctan(x)+c}}$

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