Mathematics

Write a value of 
$$\int { { x }^{ 2 } } \sin { { x }^{ 3 } } dx\quad $$


SOLUTION
Let $$t={x}^{3}\Rightarrow\,dt=3{x}^{2}dx$$

$$\Rightarrow\,\dfrac{dt}{3}={x}^{2}dx$$

$$I=\displaystyle\int{{x}^{2}\sin{{x}^{3}}dx}$$

$$=\displaystyle\int{\sin{t}\dfrac{dt}{3}}$$

$$=\dfrac{1}{3}\displaystyle\int{\sin{t}dt}$$

$$=\dfrac{1}{3}\left(-\cos{t}\right)+c$$ since $$\displaystyle\int{\sin{x}dx}=-\cos{x}+c$$

$$=-\dfrac{1}{3}\cos{t}+c$$    .......where $$c$$ is the constant of integration

$$=-\dfrac{1}{3}\cos{{x}^{3}}+c$$ ........where $$t={x}^{3}$$
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Subjective Medium Published on 17th 09, 2020
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