Answer

Verified

389.7k+ views

**Hint:**The given integral $\sin \left( {{x}^{\dfrac{1}{2}}} \right)dx$ has \[{{x}^{\dfrac{1}{2}}}\] or \[\sqrt{x}\] as an argument to the sine function, which is making it complex. So we will simplify the integral by substituting \[{{x}^{\dfrac{1}{2}}}\] equal to some variable, say $t$. On making this substitution our integral will become simplified. Then we have to use the by-parts method to solve the integral obtained. Finally, we have to back substitute $t$ to \[{{x}^{\dfrac{1}{2}}}\] to get the final integral.

**Complete step-by-step answer:**

The integral given in the above question is

\[I=\int{\sin \left( {{x}^{\dfrac{1}{2}}} \right)dx}..........(i)\]

As can be seen above, the square root function \[{{x}^{\dfrac{1}{2}}}\] as an argument to the sine function is making the integral complex. So we first have to simplify the above integral by removing the square root function by substituting it to some variable $t$, that is,

$\Rightarrow t={{x}^{\dfrac{1}{2}}}.........(ii)$

Differentiating both sides with respect to $x$, we have

$\Rightarrow \dfrac{dt}{dx}=\dfrac{d\left( {{x}^{\dfrac{1}{2}}} \right)}{dx}$

Now, we know that the differentiation of ${{x}^{n}}$ is equal to $n{{x}^{n-1}}$. So the above equation becomes

\[\begin{align}

& \Rightarrow \dfrac{dt}{dx}=\dfrac{1}{2}{{x}^{\dfrac{1}{2}-1}} \\

& \Rightarrow \dfrac{dt}{dx}=\dfrac{1}{2}{{x}^{-\dfrac{1}{2}}} \\

& \Rightarrow \dfrac{dt}{dx}=\dfrac{1}{2{{x}^{\dfrac{1}{2}}}} \\

\end{align}\]

Substituting (ii) in the above equation, we get

\[\Rightarrow \dfrac{dt}{dx}=\dfrac{1}{2t}\]

By cross multiplying, we can write

\[\begin{align}

& \Rightarrow 2tdt=dx \\

& \Rightarrow dx=2tdt.........(iii) \\

\end{align}\]

Substituting (ii) and (iii) in (i), we get

$\begin{align}

& \Rightarrow I=\int{\sin t\left( 2tdt \right)} \\

& \Rightarrow I=\int{2t\sin tdt}......(iv) \\

\end{align}$

Now, we will use the by parts method to solve the above integral. From the by parts method, we know that

$\int{f\left( t \right)g\left( t \right)dt}=f\left( t \right)\int{g\left( t \right)dt}-\int{f'\left( t \right)\left( \int{g\left( t \right)dt} \right)dt}$

Choosing $f\left( t \right)=2t$ and $g\left( t \right)=\sin t$, the integral in (iv) can be written as

\[\begin{align}

& \Rightarrow I=2t\int{\sin tdt}-\int{\dfrac{d\left( 2t \right)}{dt}\left( \int{\sin tdt} \right)dt} \\

& \Rightarrow I=2t\int{\sin tdt}-\int{2\left( \int{\sin tdt} \right)dt} \\

\end{align}\]

We know that \[\int{\sin tdt}=-\cos t\]. Putting it in the above integral, we get

\[\begin{align}

& \Rightarrow I=2t\left( -\cos t \right)-\int{2\left( -\cos t \right)dt} \\

& \Rightarrow I=-2t\cos t+2\int{\cos tdt} \\

\end{align}\]

Now, we know that \[\int{\cos tdt}=\sin t\]. Putting it above, we get

\[\begin{align}

& \Rightarrow I=-2t\cos t+2\sin t+C \\

& \Rightarrow I=2\sin t-2t\cos t+C \\

\end{align}\]

Finally, substituting (ii) in the above equation, we get

\[\begin{align}

& \Rightarrow I=2\sin {{x}^{\dfrac{1}{2}}}-2{{x}^{\dfrac{1}{2}}}\cos {{x}^{\dfrac{1}{2}}}+C \\

& \Rightarrow I=2\left( \sin {{x}^{\dfrac{1}{2}}}-{{x}^{\dfrac{1}{2}}}\cos {{x}^{\dfrac{1}{2}}} \right)+C \\

\end{align}\]

Hence, the integral of $\sin \left( {{x}^{\dfrac{1}{2}}} \right)dx$ is equal to \[2\left( \sin {{x}^{\dfrac{1}{2}}}-{{x}^{\dfrac{1}{2}}}\cos {{x}^{\dfrac{1}{2}}} \right)+C\].

**Note:**Do not forget to add a constant after the integration since we have solved an indefinite integral. Also, while substituting ${{x}^{\dfrac{1}{2}}}=t$, do not replace $dx$ by $dt$ directly. We have to take the differential on both sides of the equation ${{x}^{\dfrac{1}{2}}}=t$ for obtaining $dt$ in terms of $dx$.

Recently Updated Pages

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

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

Advantages and disadvantages of science

10 examples of friction in our daily life

Trending doubts

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

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

Which are the Top 10 Largest Countries of the World?

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

10 examples of law on inertia in our daily life

Write a letter to the principal requesting him to grant class 10 english CBSE

Difference Between Plant Cell and Animal Cell

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