Answer

Verified

381.6k+ views

**Hint:**Here, we will use the Integration by Parts formula to simplify the integrand. Then by using the suitable Integral formula, we will find the integral of the given function. Integration is defined as the summation of all the discrete data.

**Formula Used:**

We will use the following formula:

1. Derivative formula: \[\dfrac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}\], \[\dfrac{d}{{dx}}\left( C \right) = 1\]

2. Integration by Parts: \[\int {uvdx = uv - \int {vdu} } \]

3. Integral Formula: \[\int {\cos tdt = \sin t} \], \[\int {\sin tdt = - \cos t} \]

**Complete Step by Step Solution:**

We are given an integral function \[\int {\cos \sqrt x dx} \] .

Let the given integral function be \[I\]

\[I = \int {\cos \sqrt x dx} \] …………………………………………….\[\left( 1 \right)\]

Now, we will substitute a variable for the radical expression in the integrand, we get

\[t = \sqrt x = {\left( x \right)^{\dfrac{1}{2}}}\] ……………………………………...\[\left( 2 \right)\]

Now, we will differentiate the variable with respect to \[x\] using the derivative formula \[\dfrac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}\], we get

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

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

Now, by rewriting the equation, we get

\[ \Rightarrow dx = 2\sqrt x dt\]

\[ \Rightarrow dx = 2tdt\] ……………………………………………\[\left( 3 \right)\]

Substituting the equation \[\left( 2 \right)\] and \[\left( 3 \right)\] in equation \[\left( 1 \right)\] , we get

\[I = \int {\cos t \cdot 2tdt} \]

\[ \Rightarrow I = 2\int {t\cos tdt} \]

Now, by using Integration by Parts formula \[\int {uvdx = uv - \int {vdu} } \] for the Integral function, we get \[u = t\] according to ILATE rule and \[v = \cos t\] .

Now, we will differentiate the variable \[u\], so we get

\[du = dt\]

Now, we will integrate the variable \[v\] using the formula \[\int {\cos tdt = \sin t} \], so we get

\[\int {\cos tdt = \sin t} \]

Substituting differentiated variable and integrated variable in the integration by parts formula, we get

\[ \Rightarrow \int {t\cos tdt = t\sin t - \int {\sin tdt} } \]

Now, by using the integral formula \[\int {\sin tdt = - \cos t} \], we get

\[ \Rightarrow \int {t\cos tdt = t\sin t - \left( { - \cos t} \right)} \]

\[ \Rightarrow \int {t\cos tdt = t\sin t + \cos t} \] …………………………………………\[\left( 4 \right)\]

Now, by substituting \[I = 2\int {t\cos tdt} \] in the equation \[\left( 4 \right)\], we get

\[ \Rightarrow I = 2\left[ {t\sin t + \cos t} \right] + c\]

Now, by substituting the equation \[\left( 2 \right)\], we get

\[ \Rightarrow I = 2\left[ {\sqrt x \sin \sqrt x + \cos \sqrt x } \right]\]

Therefore, the value of \[\int {\cos \sqrt x dx} \] is \[2\left[ {\sqrt x \sin \sqrt x + \cos \sqrt x } \right]\].

**Thus, option (D) is the correct answer.**

**Note:**

We know that Integration is the process of adding small parts to find the whole parts. While performing the Integration by Parts, the first function is selected according to ILATE rule where Inverse Trigonometric function, followed by Logarithmic function, Arithmetic Function, Trigonometric Function and at last Exponential Function. Integration by Parts is applicable only when the integrand is a product of two Functions.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Why Are Noble Gases NonReactive class 11 chemistry CBSE

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

How many crores make 10 million class 7 maths CBSE

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

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

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

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

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