Evaluate the following integrals: ^ 2x (- +2 )dx - Vidyadip

Question

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

✓

Answer

Let $\text{I}=\int\text{e}^{2\text{x}}(-\sin\text{x}+2\cos\text{x})\text{dx}$
$=-\int\text{e}^{2\text{x}}\sin\text{x dx}+2\int\text{e}^{2\text{x}}\cos\text{x dx}$
Applying by parts in the 2^nd integrand
$\therefore\text{I}=-\int\text{e}^{2\text{x}}\sin\text{x dx}+2\Big\{\frac{1}{2}\text{e}^{2\text{x}}\cos\text{x}+\int\frac{1}{2}\text{e}^{2\text{x}}\sin\text{x dx}\Big\}$
$=-\int\text{e}^{2\text{x}}\sin\text{x dx}+\text{e}^{2\text{x}}\cos\text{x}+\int\text{e}^{2\text{x}}\sin\text{x dx}+\text{C}$
$=\text{e}^{2\text{x}}\cos\text{x}+\text{C}$
Thus,
$\text{I}=\text{e}^{2\text{x}}\cos\text{x + C}$

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 "4 Marks" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

If for function $\phi(\text{x})=\lambda\text{x}^2+7\text{x}-4, \phi(5)=97,$ find $\lambda.$

The sum of three numbers is 2. If twice the second number is added to the sum of first and third, the sum is 1. By adding second and third number to five times the first number, we get 6. Find the three numbers by using matrices.

Evaluate the following integrals:
$\int\sqrt{2\text{x}^2+3\text{x}+4}\text{dx}$

A and B toss a coin alternately till one of them gets a head and wins the game. If A starts the game, find the probability that B will win the game.

Without expanding, show that the values of the following determinant are zero:
$\begin{vmatrix}1&43&6\\7&35&4\\3&17&2\end{vmatrix}$

Find the equations of the tangent and the normal to the following curves at the indicated points.
$\text{x}^{\frac{2}{3}}+\text{y}^{\frac{2}{3}}=2\text{ at }(1,1)$

Differentiate the following functions with respect to x:
$(\log\text{x})^{\log\text{x}}$

Find the equation of the plane which contains the line of intersection of the planes $\overrightarrow{\text{r}}\cdot(\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}})-4=0,\overrightarrow{\text{r}}\cdot(\hat{\text{2i}}+\hat{\text{j}}-\hat{\text{k}})+5=0$and which is perpendicular to the plane$\overrightarrow{\text{r}}\cdot(5\hat{\text{i}}+3\hat{\text{j}}-6\hat{\text{k}})+8=0.$

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}+\text{y}\cot\text{x}=\text{x}^2\cot\text{x}+2\text{x}$

Find the solution of diffrential equation $\left(\tan ^{-1} y-x\right) d y=\left(1+y^2\right) d x$, when $x=0, y=0$.