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Question Number 141085 by iloveisrael last updated on 15/May/21

Answered by mindispower last updated on 15/May/21

sin(3a)=3sin(a)−4sin^3 (a)......

$${sin}\left(\mathrm{3}{a}\right)=\mathrm{3}{sin}\left({a}\right)−\mathrm{4}{sin}^{\mathrm{3}} \left({a}\right)...... \\ $$

Answered by qaz last updated on 15/May/21

∫((sin (((3x)/4)))/(sin ((x/4))))dx=∫((3sin ((x/4))−4sin^3 ((x/4)))/(sin ((x/4))))dx  =3x+2∫(1−2sin^2 (x/4)−1)dx  =x+2∫cos (x/2)dx  =x+4sin (x/2)+C

$$\int\frac{\mathrm{sin}\:\left(\frac{\mathrm{3}{x}}{\mathrm{4}}\right)}{\mathrm{sin}\:\left(\frac{{x}}{\mathrm{4}}\right)}{dx}=\int\frac{\mathrm{3sin}\:\left(\frac{{x}}{\mathrm{4}}\right)−\mathrm{4sin}\:^{\mathrm{3}} \left(\frac{{x}}{\mathrm{4}}\right)}{\mathrm{sin}\:\left(\frac{{x}}{\mathrm{4}}\right)}{dx} \\ $$$$=\mathrm{3}{x}+\mathrm{2}\int\left(\mathrm{1}−\mathrm{2sin}\:^{\mathrm{2}} \frac{{x}}{\mathrm{4}}−\mathrm{1}\right){dx} \\ $$$$={x}+\mathrm{2}\int\mathrm{cos}\:\frac{{x}}{\mathrm{2}}{dx} \\ $$$$={x}+\mathrm{4sin}\:\frac{{x}}{\mathrm{2}}+{C} \\ $$

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