Evaluate the following integrals: ^ 2x (3x+4)dx - Vidyadip

Question

Evaluate the following integrals:
$\int\text{e}^{2\text{x}}\cos(3\text{x}+4)\text{dx}$

✓

Answer

Let $\text{I}=\int\text{e}^{2\text{x}}\cos(3\text{x}+4)\text{dx}$
Integrating by parts
$\text{I}=\text{e}^{2\text{x}}\frac{\sin(3\text{x}+4)}{3}-\int2\text{e}^{2\text{x}}\frac{\sin(3\text{x}+4)}{3}\text{dx}$
$=\frac{1}{3}\text{e}^{2\text{x}}\sin(3\text{x}+4)-\frac{2}{3}\int\text{e}^{2\text{x}}\sin(3\text{x}+4)\text{dx}$
$=\frac{1}{3}\text{e}^{2\text{x}}\sin(3\text{x}+4)-\frac{2}{3}\Big[-\text{e}^{2\text{x}}\frac{\cos(3\text{x}+4)}{3}+\int2\text{e}^{2\text{x}}\frac{\cos(3\text{x}+4)}{3}\text{dx}\Big]+\text{C}$
$\text{I}=\frac{\text{e}^{2\text{x}}}{9}\{2\cos(3\text{x}+4)+3\sin(3\text{x}+4)\}+\text{C}$
Hence,
$\text{I}=\frac{\text{e}^{2\text{x}}}{13}\{2\cos(3\text{x}+4)+3\sin(3\text{x}+4)\}+\text{C}$

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 "5 Marks Questions" 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

Find the equations of the tangent and the normal to the following curves at the indicated points.
$\text{y}^2=4\text{x}\text{ at }(1,2)$

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

If $\text{A}=\begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix},$ find $A^{-1}$ and prove that $A^2 - 4A - 5I = 0$.

Find the angle between the follwing pairs of lines:$\vec{\text{r}}=\big(3\hat{\text{i}}+2\hat{\text{j}}-4\hat{\text{k}}\big)+\lambda\big(\hat{\text{i}}+2\hat{\text{j}}+2\hat{\text{k}}\big)$ and $\vec{\text{r}}=\big(5\hat{\text{j}}-2\hat{\text{k}}\big)+\mu\big(3\hat{\text{i}}+2\hat{\text{j}}+6\hat{\text{k}}\big)$

Find the absolute maximum and absolute minimum values of the function f given by $\text{f(x)} = \sin^{2}\text{x} - \cos\text{x , x}\in [0, \pi].$

Show that the lines $\vec{r}=(\hat{i}+2 \hat{j}+3 \hat{k})+\lambda(2 \hat{i}+3 \hat{j}+4 \hat{k})$ and $\vec{r}=(4 \hat{i}+\hat{j})+\mu(5 \hat{i}+2 \hat{j}+\hat{k})$ intersect. Also, find their point intersection.

Form the differential equation of all circles which pass through origin and whose centres lie on Y-axis.

If f(2a - x) = -f(x), prove that $\int\limits^{2\text{a}}_0\text{f(x)}\text{dx}=0$

Evaluate the following definite integral as limit of sum:
$\int\limits_{1}^{4}(\text{x}^{2}-\text{x)}\ \text{dx}$

There are two types of fertilisers 'A' and 'B'. 'A' consists of 12 % nitrogen and 5 % phosphoric acid whereas 'B' consists of 4 % nitrogen and 5 % phosphoric acid. After testing the soil conditions, farmer finds that he needs at least 12 kg of nitrogen and 12 kg of phosphoric acid for his crops. If 'A' costs 10 per kg and 'B' cost 8 per kg, then graphically determine how much of each type of fertiliser should be used so that nutrient requirements are met at a minimum cost.