Question

Calcule a seguinte integral:

Answer 1

Redução de integral para o seno :

$\displaystyle \int \text{sen}^{\text n}(\text x)\text{dx} = \frac{-\text{cos(x)}.\text{sen}^{\text n-1}(\text x)}{\text n}+\frac{\text n+1}{\text n}\int \text{sen}^{\text n-2}(\text x)\text {dx}$

Temos :

$\displaystyle \int \text{sen}^{4}(\text {x)dx} = \frac{-\text{cos(x)sen}^{4-1}(\text {x)}}{4}+\frac{4-1}{4}\int \text{sen}^{4-2}(\text x)\text{dx} \\\\\\ \frac{-\text{cos(x).sen}^3(\text x)}{4}+\frac{3}{4}.\int \text{sen}^2(\text x)\text{dx}$

Vamos calcular a integral de sen²(x) :

$\displaystyle \text{cos(2x)}=1-2.\text{sen}^2(\text x) \\\\\ \text{sen}^2(\text x)=\frac{1-\text{cos(2x)}}{2} \\\\ \therefore \\\\ \int \text{sen}^2(\text x)\text{dx} = \int \frac{[1-\text{cos(2x)]dx}}{2} \\\\\\ \frac{1}{2}[\int 1\text {dx} - \int \text{cos(2x)dx}\ ] \\\\\\ \frac{\text x}{2}-\frac{1}{2}\frac{\text{sen(2x)}}{2} = \frac{\text x}{2}-\frac{\text{sen(2x)}}{4}$

Voltando de onde paramos :

$\displaystyle \frac{-\text{cos(x).sen}^3(\text x)}{4}+\frac{3}{4}.\int \text{sen}^2(\text x)\text{dx}\\\\\\ \frac{-\text{cos(x).sen}^3(\text x)}{4} + \frac{3}{4}.[ \ \frac{\text x}{2} -\frac{\text{sen(2x)}}{4}\ ]+ \text C \\\\\\ \huge\boxed{\frac{-\text{cos(x).sen}^3(\text x)}{4} + \frac{3\text x}{8} -\frac{3.\text{sen(2x)}}{16}+ \text C }$

Letra E