Question

Calculați :

$1) \int {sin}^{2} x \: dx$

$2) \int{cos}^{2} x \: dx$
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Answer 1

Răspuns:ai raspunsul in poza atasata. Scuze pentru scrisul urat :).

P.S. pentru integrala din cos(ax) exista formula ....asa cum am precizat si in poza .

Explicație pas cu pas:

Answer 2

$\displaystyle I_1 = \int \sin^2 x\, dx \\ I_2 = \int \cos^2 x\, dx \\ \\ I_1+I_2 = \int(\sin^2 x+\cos^2 x)\, dx = \int 1 \, dx = x+C\\ I_2-I_1 = \int(\cos^2 x-\sin^2 x)\, dx = \int \cos(2x)\, dx = \\ \\ = \dfrac{1}{2}\int \Big(\sin(2x)\Big)'\, dx =\dfrac{\sin(2x)}{2}+C\\ \\\\ \text{Adunam cele 2 relatii:}\\ \\ \Rightarrow 2I_2 = x+\dfrac{\sin(2x)}{2}+C \Rightarrow \boxed{I_2 = \dfrac{x}{2}+\dfrac{\sin(2x)}{4}+C}$

$\Rightarrow I_1 = x - I_2+C \Rightarrow \boxed{I_1 = \dfrac{x}{2} - \dfrac{\sin(2x)}{4}+C}$

$\Rightarrow \displaystyle \int \sin^2 x\, dx= \dfrac{x}{2} - \dfrac{\sin(2x)}{4}+C \\ \\\Rightarrow \int \cos^2 x\, dx= \dfrac{x}{2} + \dfrac{\sin(2x)}{4}+C$