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Question: What is the integral of \(\int{{{\sin }^{2}}\left( x \right).{{\cos }^{2}}\left( x \right)dx}\) ?...

What is the integral of ∫sin⁡2(x).cos⁡2(x)dx\int{{{\sin }^{2}}\left( x \right).{{\cos }^{2}}\left( x \right)dx} ?

Explanation

Solution

To solve the given integral we will use properties of trigonometric functions and the formula to integrate the trigonometric functions. Firstly as we know that sin⁡(2x)=2sin⁡xcos⁡x\sin \left( 2x \right)=2\sin x\cos x so we will multiply and divide the equation by 4 then we will obtain our equation in terms of sin⁡(2x)\sin \left( 2x \right) power. Next we will use the formula of sin⁡(2x)\sin \left( 2x \right) in terms of cosine. Finally we will integrate the equation and get the desired answer.

Complete step by step solution:
We have to integrate the equation:
∫sin⁡2(x).cos⁡2(x)dx\int{{{\sin }^{2}}\left( x \right).{{\cos }^{2}}\left( x \right)dx}…..(1)\left( 1 \right)
So as we know sin⁡(2x)=2sin⁡xcos⁡x\sin \left( 2x \right)=2\sin x\cos x we will multiply and divide the above equation by 2 and simplify to get,
=∫44×sin⁡2(x).cos⁡2(x)dx =14∫4.sin⁡2(x).cos⁡2(x)dx \begin{aligned} & = \int{\dfrac{4}{4}\times {{\sin }^{2}}\left( x \right).{{\cos }^{2}}\left( x \right)dx} \\\ & = \dfrac{1}{4}\int{4.{{\sin }^{2}}\left( x \right).{{\cos }^{2}}\left( x \right)dx} \\\ \end{aligned}
Using formula in equation (1) on above equation we get,
=14∫(4sin⁡2(x).cos⁡2(x)) =14∫(2sin⁡(x).cos⁡(x))2 =14∫(sin⁡(2x))2 \begin{aligned} & = \dfrac{1}{4}\int{\left( 4{{\sin }^{2}}\left( x \right).{{\cos }^{2}}\left( x \right) \right)} \\\ & = \dfrac{1}{4}\int{{{\left( 2\sin \left( x \right).\cos \left( x \right) \right)}^{2}}} \\\ & = \dfrac{1}{4}\int{{{\left( \sin \left( 2x \right) \right)}^{2}}} \\\ \end{aligned}
Next, we will use the identity 2sin⁡2(2x)=1−cos⁡(4x)2{{\sin }^{2}}\left( 2x \right)=1-\cos \left( 4x \right) in above equation and get,
=14∫1−cos⁡(4x)2dx =14∫12−cos⁡(4x)2dx \begin{aligned} & = \dfrac{1}{4}\int{\dfrac{1-\cos \left( 4x \right)}{2}dx} \\\ & = \dfrac{1}{4}\int{\dfrac{1}{2}-\dfrac{\cos \left( 4x \right)}{2}}dx \\\ \end{aligned}
On integrating the above equation we get,
=14(x2−sin⁡(4x)4×2)+c =x8−sin⁡(4x)32+c \begin{aligned} & = \dfrac{1}{4}\left( \dfrac{x}{2}-\dfrac{\sin \left( 4x \right)}{4\times 2} \right)+c \\\ & = \dfrac{x}{8}-\dfrac{\sin \left( 4x \right)}{32}+c \\\ \end{aligned}
So we get the answer as x8−sin⁡(4x)32+c\dfrac{x}{8}-\dfrac{\sin \left( 4x \right)}{32}+c
Hence, integral of ∫sin⁡2(x).cos⁡2(x)dx\int{{{\sin }^{2}}\left( x \right).{{\cos }^{2}}\left( x \right)dx} is x8−sin⁡(4x)32+c\dfrac{x}{8}-\dfrac{\sin \left( 4x \right)}{32}+c

Note: The term integration is generally used to find the area, volume and many other things that are useful in mathematics. It is similar to adding a slice to make a whole. If the upper and lower limit of the integral is not present i.e. it is an indefinite integrals we use the constant cc to the final answer. Integration is opposite of differentiation as we know differentiation of sine is cosine but integration of cosine is sine. Integration is also known as antiderivative. Real life examples where we use integration is to calculate the centre of mass or gravity; it also is very helpful in predicting the position of the planets.