Evaluate the following integrals: ^ 2x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\text{e}^{2\text{x}}\sin\text{x}\cos\text{x }\text{dx}$

✓

Answer

Let $\text{I}=\int\text{e}^{2\text{x}}\sin\text{x}\cos\text{x }\text{dx}$
$=\frac{1}{2}\int\text{e}^{2\text{x}}2\sin\text{x}\cos\text{x dx}$
$=\frac{1}{2}\int\text{e}^{2\text{x}}\sin2\text{x dx}$
We know that
$\int\text{e}^{2\text{x}}\sin\text{bx dx}=\frac{\text{e}^{2\text{x}}}{\text{a}^2+\text{b}^2}\{\text{a}\sin\text{bx}-\text{b}\cos\text{bx}\}+\text{C}$
$\Rightarrow\int\text{e}^{2\text{x}}\sin\text{2x dx}=\frac{\text{e}^{2\text{x}}}{8}\{2\sin2\text{x}-2\cos2\text{x}\}+\text{C}$
$\therefore\ \text{I}=\frac{1}{2}.\frac{\text{e}^{2\text{x}}}{8}\{2\sin2\text{x}-2\cos2\text{x}\}+\text{C}$
$\therefore\ \text{I}=\frac{\text{e}^{2\text{x}}}{8}\{\sin2\text{x}-\cos2\text{x}\}+\text{C}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "Solve the Following Question.(5 Marks)" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Find the nth derivative of the following:

$y=e^{8 x} \cdot \cos (6 x+7)$

Three cards are cdrawn successively with replacement from a well shffled deck of 52 cards. A random variable X denotes the number of hearts in the three cards drawn. Determine the probability distribution of X.

If $\text{f(x)}=\log\Big\{\frac{\text{u(x)}}{\text{v(x)}}\Big\},\text{u}(1)=\text{v}(1)$ and u'(1) = v'(1) = 2, then find the value of f(1).

In $\triangle ABC$ prove that $\cot \frac{A}{2}+\cot \frac{B}{2}+\cot \frac{C}{2}=\frac{a+b+c}{b+c-a} \cot \frac{A}{2}$

Solve the following initial value problems:
$\text{x}\frac{\text{dy}}{\text{dx}}-\text{y + x}\sin\Big(\frac{\text{y}}{\text{x}}\Big)=0,\text{y}(2)=\text{x}$

Evaluate the following intregals:
$\int\frac{\text{x}^2}{(\text{x}-1)(\text{x}-2)(\text{x}-3)}\ \text{dx}$

There are three categories of students in a class of 60 students:

A : Very hardworking

B : Regular but not so hardworking

C : Careless and irregular 10 students are in category A, 30 in category B and the rest in category C.

It is found that the probability of students of category A, unable to get good marks in the final year examination is 0.002, of category B it is 0.02 and of category C, this probability is 0.20. A student selected at random was found to be one who could not get good marks in the examination. Find the probability that this student is category C.

Evaluate the following integrals:
$\int\text{e}^{2\text{x}}\Big(\frac{1-\sin2\text{x}}{1-\cos2\text{x}}\Big)\text{dx}$

Determine the equation of the line passing through the points (1, 2, -4) and perpendicular to the lines $\frac{\text{x}-8}{8}=\frac{\text{y}+9}{-16}=\frac{\text{z}-10}{7}$ and $\frac{\text{x}-15}{3}=\frac{\text{y}-29}{8}=\frac{\text{z}-5}{-5}.$

Solve the following differential equations:$\tan\text{y}\frac{\text{dy}}{\text{dx}}=\sin(\text{x}+\text{y})+\sin(\text{x}-\text{y})$