Answer

Verified

387.6k+ views

**Hint:**In the above type of integration question first of all we will have to convert them by using trigonometric formulae in that form in which we can easily integrate them, so, we have to remember the sine and cosine sum angle formulae. After that use integration by substitution method to solve the problem.

**Complete step by step solution:**

In the above question, we have to find the integral of ${\sin ^3}x{\cos ^3}x$ which is in the multiplication form and we don’t know the integration of this kind. So, we will try to split it by the formula,

${\cos ^2}x = 1 - {\sin ^2}x$

So, by using the above formulae we can write the given expression is as follow;

$ \Rightarrow \int {{{\sin }^3}x{{\cos }^3}xdx} = \int {{{\sin }^3}x\left( {1 - {{\sin }^2}x} \right)\cos xdx} $

There are times when the given function is a little complicated and thus, making it difficult for us to integrate. To make it easy we use a different independent variable to make it easier to integrate. This is known as integration by substitution.

Now apply integration by substitution to integrate it.

Let us assume $\sin x = t$,

Differentiate it with respect to $x$,

$ \Rightarrow \cos x = \dfrac{{dt}}{{dx}}$

Cross multiply the terms,

$ \Rightarrow \cos xdx = dt$

Substitute the value in the equation,

$ \Rightarrow \int {{{\sin }^3}x{{\cos }^3}xdx} = \int {{t^3}\left( {1 - {t^2}} \right)dt} $

Open the brackets and multiply the terms,

$ \Rightarrow \int {{{\sin }^3}x{{\cos }^3}xdx} = \int {\left( {{t^3} - {t^5}} \right)dt} $

Integrate the terms,

$ \Rightarrow \int {{{\sin }^3}x{{\cos }^3}xdx} = \dfrac{{{t^4}}}{4} - \dfrac{{{t^6}}}{6} + c$

Now substitute back the value of $t$ in the equation,

$ \Rightarrow \int {{{\sin }^3}x{{\cos }^3}xdx} = \dfrac{{{{\sin }^4}x}}{4} - \dfrac{{{{\sin }^6}x}}{6} + c$

**Hence, the integral of the given function in the above question will be $\dfrac{{{{\sin }^4}x}}{4} - \dfrac{{{{\sin }^6}x}}{6} + c$.**

**Note:**Integration is a way of adding slices to find the whole. Integration can be used to find areas, volumes, central points and many useful things. But it is easiest to start with finding the area under the curve of a function.

In such types of questions always choose substitution which makes integration simple, in above integration we choose $\sin x = t$, so it makes integration simple, then we easily integrate using some basic property of integration which is stated above, then simplify we will get the required answer.

Recently Updated Pages

What number is 20 of 400 class 8 maths CBSE

Which one of the following numbers is completely divisible class 8 maths CBSE

What number is 78 of 50 A 32 B 35 C 36 D 39 E 41 class 8 maths CBSE

How many integers are there between 10 and 2 and how class 8 maths CBSE

The 3 is what percent of 12 class 8 maths CBSE

Find the circumference of the circle having radius class 8 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

One cusec is equal to how many liters class 8 maths CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE

Change the following sentences into negative and interrogative class 10 english CBSE