Question

How to use u substitution for $\sin 2x$?

Answer 1

According to the question given in the question we have to find the substitution of $\sin 2x$ using u substitution method. So, first of all according to u substitution method we have to substitute any of the term so that we can easily integrate or differentiate the given function and according to this question to integrate the function which is $\sin 2x$ we have to substitute $2x$ as u means we have to let $2x$ as u then we have to differentiate it.
Now, to differentiate the expression obtained we have to use the formula to determine the differentiation which is as mentioned below:

Formula used: $\Rightarrow \dfrac{d}{{dy}}ay = a............(A)$
Hence, with the help of the formula (A) above, we can determine the differentiation of the expression we let.
Now, we have to substitute all the values obtained after the differentiation in the integration so that we can easily determine the values of the integration.
Now, to solve the integration we have to use the formula which is as mentioned below:
$\Rightarrow \int {\sin xdx = - \cos x + C..............(B)}$
Where, C is the constant term.
Now, we have to substitute the value of u as we have let in the starting steps of the solution so that we can determine the integration of the given function.

Complete step-by-step solution:
Step 1: First of all according to u substitution method we have to substitute any of the term so that we can easily integrate or differentiate the given function and according to this question to integrate the function which is $\sin 2x$ we have to substitute $2x$ as u means we have to let $2x$ as u then we have to differentiate it. Hence,
$\Rightarrow$let $u = 2x$………….(1)
Step 2: Now, to differentiate the expression obtained we have to use the formula (A) to determine the differentiation which is as mentioned in the solution hint. Hence,
$\Rightarrow \dfrac{{du}}{{dx}} = \dfrac{{d2x}}{{dx}} \\\ \Rightarrow du = 2dx$
Now, rearranging the terms obtained just above,
$\Rightarrow dx = \dfrac{{du}}{2}.............(2)$
Step 3: Now, we have to substitute all the values obtained after the differentiation in the integration so that we can easily determine the values of the integration. Hence,
$\Rightarrow I = \dfrac{1}{2}\int {\sin udu...........(3)}$
Step 4: Now, to solve the integration (3) as obtained in the solution step 3 we have to use the formula (A) which is as mentioned in the solution hint. Hence,
$\Rightarrow I = \dfrac{1}{2}( - \cos u) + C \\\ \Rightarrow I = - \dfrac{1}{2}\cos u + C..............(4)$
Step 5: Now, we have to substitute the value of u in the integration (4), as we have let in the starting steps of the solution so that we can determine the integration of the given function. Hence,
$\Rightarrow - \dfrac{1}{2}\cos 2x + C$

Hence, with the help of formula (A) we have determined the integration of $\sin 2x$ to use the substitution method which is $- \dfrac{1}{2}\cos 2x + C$.

Note: To find the integration by u substitution method it is necessary that we have to let the term $2x$ as a constant term and then we have to find the differentiation of the term we let.
It is necessary that we have to substitute the term we let as u after obtaining the integration of the function.