Download App

INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS3 Marks
Question
Evaluate the following integrals:
$\int\text{e}^{\text{x}}(\cos\text{x}-\sin\text{x})\text{dx}$
✓

Answer

Let $\text{I}=\int\text{e}^{\text{x}}(\cos\text{x}-\sin\text{x})\text{dx}$
$=\int\text{e}^{\text{x}}\cos\text{x dx}-\int\text{e}^{\text{x}}\sin\text{x dx}$
Integrating by parts
$=\text{e}^{\text{x}}\cos\text{x}-\int\text{e}^{\text{x}}\Big(\frac{\text{d}}{\text{dx}}\cos\text{x}\Big)\text{dx}-\int\text{e}^{\text{x}}\sin\text{x dx}$
$=\text{e}^{\text{x}}\cos\text{x}+\int\text{e}^{\text{x}}\sin\text{x dx}-\int\text{e}^{\text{x}}\sin\text{x dx}$
$=\text{e}^{\text{x}}\cos\text{x}+\text{C}$
$\therefore\int\text{e}^{\text{x}}(\cos\text{x}-\sin\text{x})\text{dx}=\text{e}^{\text{x}}\cos\text{x}+\text{C}$

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 "3 Marks" from INTEGRALS→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 value of $\int \frac{\mathrm{dx}}{\mathrm{e}^{\mathrm{x}}-1}$.
→
Show that f: [–1, 1] →R, given by $f(\text{x})=\frac{\text{x}}{(\text{x}+2)}$ is one-one. Find the inverse of the function f:[-1, 1] → Range f.
(Hint: For $\text{y}\in\text{Range f},\text{y}=f(\text{x})=\frac{\text{x}}{\text{x}+2},$ for some x in [-1, 1], i.e., $\text{x}=\frac{2\text{y}}{(1-\text{y})}$)
→
If y = loge x, then find $\triangle\text{y}$ when x = 3 and $\triangle\text{x}$  = 0.03.
→
Solve the following equation for x:
$\tan^{-1}\Big(\frac{2\text{x}}{1-\text{x}^2}\Big)+\cot^{-1}\Big(\frac{1-\text{x}^2}{2\text{x}}\Big)=\frac{2\pi}{3},\text{x}>0$
→
Evaluate the following integrals:
$\int\frac{1}{\sqrt{1-\text{x}^2}(\sin^{-1}\text{x})^2}\text{dx}$
→
For a loaded die, the probabilities of outcome are given as under:
P(1) = P(2) = 0.2, P(3) = P(5) = P(6) = 0.1 and P(4) = 0.3. the die is thrown two times. If the die were fair, determine whether or not the events A and B are independent.
→
Let A = {1, 2, 3}, and let R3 = {(1, 3), (3, 3)}. Find whether or not the relations R3 on A is:
  1. Reflexive.
  2. Symmetric.
  3. Transitive.
→
Prove that the greatest integer function defined by $f\left( x \right) = \left[ x \right],0 < x < 3$ is not differentiable at x = 1 and x = 2.
→
Evaluate the following integrals:
$\int^\limits\frac{\pi}{4}_{0}\sin^32\text{t}\cos2\text{t}\text{ dt}$
→
Using the properties of determinants, prove that

$ \begin{vmatrix} \text{a + b} & \text{b + c} & \text{c + a} \\ \text{b + c} & \text{c + a} & \text{a + b} \\ \text{c + a} & \text{a + b} & \text{b + c} \end{vmatrix}=2 \begin{vmatrix} \text{a} & \text{b} & \text{c} \\ \text{b} & \text{c} & \text{a} \\ \text{c} & \text{a} & \text{b} \end{vmatrix}$.

→