Question

Evaluate \[\int {\sin \sqrt x } dx = \]
1) \[2\left( {\sin \sqrt x - \sqrt x .\cos \sqrt x } \right) + c\]
2) \[2\left( {\sin \sqrt x + \sqrt x .\cos \sqrt x } \right) + c\]
3) \[2\left( {\sin \sqrt x - \dfrac{1}{2}\sqrt x \cos \sqrt x } \right) + c\]
4) \[\left( {\sin \sqrt x + \dfrac{1}{2}\sqrt x \cos \sqrt x } \right) + c\]

Answer 1

Hint: In order to determine the given definite integral, we have to use the substitution method of integration by substituting \[x\]with t. then, we use the product rule for further differentiation, the formula is \[\int {udv = uv - \int {vdu} } \] to evaluate to integral to obtain the required answer.

Complete step-by-step answer:
We are given the integral \[\int {\sin \sqrt x } dx\] ----------(1)
Now, let us assume \[t = \sqrt x \] ----------(2)
Perform the first derivative of the above assumed equation, then
\[
   \Rightarrow \dfrac{{dt}}{{dx}} = \dfrac{1}{{2\sqrt x }} \\
   \Rightarrow dx = 2\sqrt x dt \;
 \]
Since, \[t = \sqrt x \]\[ \Rightarrow dx = 2tdt\] ---------(3)
Here, we use the substitution method of integration to solve the above integral
As per the assumption, we have
\[\int {2t\sin t} dt = 2\int {t\sin tdt} \]
With the the product rule \[\int {udv = uv - \int {vdu} } \] and trigonometric identity \[\sin t = - \cos t\] and \[\int {\cos t = \sin t} \]. Since, we have assumed expression x as ‘t’,
Let us assume, \[u = t \Rightarrow du = 1 \cdot dt\] and \[dv = \sin t \Rightarrow v = - \cos t\]
we can write the function as
\[2\int {t\sin t} dt = 2\left[ {t( - \cos t) - \int {( - \cos t)dt} } \right]\]
Now we can simplify the integration with the trigonometric identity\[\int {\cos t = \sin t} \], then
\[2\int {t\sin t} dt = 2\left( { - t\cos t + \sin t} \right) + c\]
We can write back the assumption value ‘t’. Since ,\[t = \sqrt x \]
\[\int {\sin \sqrt x } dx = 2\left( { - \sqrt x \cos \sqrt x + \sin \sqrt x } \right) + C\]
Simplifying further, we get
\[\int {\sin \sqrt x } dx = 2\left( {\sin \sqrt x - \sqrt x \cos \sqrt x } \right) + C\]
Where, C is constant.
Therefore, the option (1) is the correct one.
Hence, we solved the integral \[\int {\sin \sqrt x } dx = 2\left( {\sin \sqrt x - \sqrt x \cos \sqrt x } \right) + C\]
So, the correct answer is “Option 1”.

Note: 1. Use a trigonometric identity formula carefully while evaluating the integrals.
2. Different types of methods of Integration:
Integration by Substitution
Integration by parts
Integration of rational algebraic function by using partial fraction
3. Integration by Substitution: The method of evaluating the integral by reducing it to standard form by a proper substitution is called integration by substitution.
4. Product rule for integration. Formula for the two functions: u and v is mentioned above.
After evaluating this integral we substitute back the value of t.
\[\int {udv = uv - \int {vdu} } \]