Mathematics

$$\displaystyle\int \sin^2(2x+5) dx$$.


SOLUTION
We are given $$I=\displaystyle\int \sin^{2}(2x+5) dx$$
Now, we know that
$$\cos 2\theta=1-2\sin^{2}\theta$$
So, $$1-\cos 2\theta=2\sin^{2}\theta$$
$$\therefore \sin^{2}\theta=\dfrac{1-\cos 2\theta}{2}$$
$$I=\displaystyle\int \left(\dfrac{1-\cos 2(2x+5)}{2}\right)dx$$
$$=\displaystyle\int \left(\dfrac{1}{2}-\dfrac{\cos(4x+10}{2}\right)dx$$
$$=\displaystyle\int \dfrac{dx}{2}-\dfrac{1}{2} \int \cos (4x+10) dx$$
$$=\dfrac{x}{2}-\dfrac{1}{2}\dfrac{\sin (4x+10)}{4}+C$$
$$I=\dfrac{x}{2}-\dfrac{\sin (4x+10)}{8}+C$$
Where $$C$$ is any arbitrary constant
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Subjective Medium Published on 17th 09, 2020
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