Question

What is the antiderivative of ${{\left( \sin x \right)}^{3}}$?

Answer 1

For solving this question you should know about finding integration of any trigonometric function. In this question it is asked to determine the antiderivative which means integration of the function and we will divide our function into two parts and then we solve that and get our answer.

Complete step by step solution:
According to the question we have to find the antiderivative or integration of ${{\left( \sin x \right)}^{3}}$. As we know that the antiderivative of any trigonometric function will be equal to the integration of the same function. We can understand it by an example.
Example 1. Find the derivative of $y={{x}^{3}}$ and also find the antiderivative of the answer for this.
For the derivative of ${{x}^{3}}$, we differentiate it with respect of $x$:
$\begin{aligned} & \Rightarrow \dfrac{dy}{dx}=\dfrac{d}{dx}\left( {{x}^{3}} \right) \\\ & \Rightarrow \dfrac{dy}{dx}=3{{x}^{2}} \\\ \end{aligned}$
Now we take the antiderivative of $3{{x}^{2}}$. It means that we have to find the integration of $3{{x}^{2}}$. So, the antiderivative of $3{{x}^{2}}$ is,
$\begin{aligned} & =\int{3.{{x}^{2}}dx} \\\ & =3\dfrac{{{x}^{3}}}{3}+c \\\ & ={{x}^{3}} \\\ \end{aligned}$
So, it is equal to our function and it is proved that the antiderivative is the integration of that term.
So, according to our question, we have to find the antiderivative of ${{\left( \sin x \right)}^{3}}$, so we have,

& \int{{{\left( \sin x \right)}^{3}}dx}=\int{{{\sin }^{2}}x\sin x.dx} \\\ & =\int{\left( 1-{{\cos }^{2}}x \right)\sin xdx} \\\ & =\int{\sin x.dx+\int{{{\cos }^{2}}x}\left( -\sin x \right)dx} \\\ \end{aligned}$$ If ${{\cos }}x=u$ then $du=\left( -\sin x \right).dx$ $ =-\cos x+\dfrac{{{\cos }^{3}}x}{3}+c $ **So, the antiderivative of ${{\left( \sin x \right)}^{3}}$ is $$-\cos x+\dfrac{{{\cos }^{3}}x}{3}+c$$.** **Note:** During calculating the antiderivative or integration of any trigonometric function you should always use the trigonometric formulas for integration. And make sure to correct all the calculations and specially ensure the correction of the power during changing when we integrate to them.