Answer

Verified

411k+ 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

Define the following A 1 volt PD B Electric power class 12 physics CBSE

In a halfwave rectifier the rms value of the ac component class 12 physics CBSE

When light travels from a rarer to a denser medium class 12 physics CBSE

The unit of electric dipole moment is A Newton B Coulomb class 12 physics CBSE

The waveforms A and B given below are given as input class 12 physics CBSE

What is the cause of Presbyopia class 12 physics CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Mention the different categories of ministers in the class 10 social science CBSE

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

Give 10 examples for herbs , shrubs , climbers , creepers

Who is the executive head of the Municipal Corporation class 6 social science CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Which monarch called himself as the second Alexander class 10 social science CBSE

Select the word that is correctly spelled a Twelveth class 10 english CBSE

Write an application to the principal requesting five class 10 english CBSE