Question

How do you find the integral of ${\sin ^2}x\cos x$?

Answer 1

Hint: Start by substituting $u = \sin x$. Substitute the values in place of the terms to make the equation easier to solve. Then we will differentiate the term. Now we will substitute these terms in the original expression and integrate.

Complete step-by-step solution:
First we will start off by substituting $u = \sin x$. Now we differentiate this term to form a proper equation and for substituting the terms.
$
  \,\,\,u = \sin x \\
  \dfrac{{du}}{{dx}} = \cos x \\
 $
So, now we can rewrite the integral as follows:
\[
   \Rightarrow \int {{{\sin }^2}x\cos x} \\
    \Rightarrow \int {{u^2}du} \\
 \]
Now we will apply the power rule and then integrate the terms separately.
So, first we evaluate the values of the integrals.
\[
    \Rightarrow \int {{u^2}du} \\
    \Rightarrow \dfrac{1}{3}{u^3} \\
 \]
Now again re substitute the values of $u$.
\[
    \Rightarrow \dfrac{1}{3}{u^3} \\
    \Rightarrow \dfrac{1}{3}{\sin ^3}x + c \\
 \]

Hence, the integral of ${(1 + \cos x)^2}$ will be \[\dfrac{1}{3}{\sin ^3}x + c\].

Additional Information: A derivative is the rate of change of a function with respect to a variable. Derivatives are fundamental to the solution of problems in calculus and differential equations. In general, scientists observe changing systems to obtain rate of change of some variable of interest, incorporate this information into some differential equation, and use integration techniques to obtain a function that can be used to predict the behaviour of the original system under diverse conditions. The power rule allows us to find the indefinite integrals of a variety of functions like polynomials, functions involving roots, and even some rational functions.

Note: While substituting the terms make sure you are taking into account the degrees and signs of the terms as well. While applying the power rule make sure you have considered the power with their respective signs