Answer

Verified

447.3k+ views

**Hint:**First of all we will take the given equation and break it in two parts by opening the bracket with integration symbols in each. Let these two terms as${I_1}\& {I_2}$. After that solve any one term and integrate with respect to $x$, then put this value in the original equation, thus we will get the answer, and add an integrating constant c with the answer.

**Complete step-by-step answer:**We have to integrate the given term i.e.:

$\int {{e^x}\left( {\sin x + \cos x} \right)dx} $

Now we can write the given term as:

By separate both functions:

$ \Rightarrow \int {{e^x}\sin xdx + } \int {{e^x}\cos xdx} .........\left( 1 \right)$

Let the first term in equation $\left( 1 \right)$ is ${I_1}$ and second term is ${I_2}$

Where:

${I_1} = $ \[\int {{e^x}\sin xdx} \]\[\]

\[

{I_2} = \int {{e^x}\cos xdx} \\

\\

\]

Now differentiate the term ${I_2}$ with respect to $x$

By using the $uv$ rule of integration:

$dx = d\left( {uv} \right)dx = udvdx + vdudx$

We will apply this rule and integrate it:

\[

{I_2} = \int {{e^x}\cos xdx} \\

\Rightarrow {I_2} = {e^x}\int {\cos xdx - \int {\left( {\dfrac{d}{{dx}}\left( {{e^x}} \right).\int {\cos x} } \right)} } dx \\

\Rightarrow {I_2} = {e^x}\sin x - \int {{e^x}} \sin xdx \\

\\

\]

Put the value of ${I_2}$in term${I_1}$

Thus we get:

\[

{I_1} = \int {{e^x}\sin xdx + {e^x}\sin x - \int {{e^x}\sin xdx} } \\

\Rightarrow {e^x}\sin x + c \\

\\

\]

Where c is integrating constant.

**Hence the correct answer is option B.**

**Note:**For the given question we have to remember that to integrate the given equation we have to remember that to break the equation in two integrations like $\int {{e^x}\sin xdx + } \int {{e^x}\cos xdx} $

called as ${I_1}\& {I_2}$ then solve ${I_2}$ part of the equation and put this value in equation $1$ and add an integrating constant c with this and this is our answer.

Recently Updated Pages

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

Mark and label the given geoinformation on the outline class 11 social science CBSE

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

Trending doubts

Difference Between Plant Cell and Animal Cell

Which are the Top 10 Largest Countries of the World?

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

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

How do you graph the function fx 4x class 9 maths CBSE

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

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