Mathematics

# Evaluate $\displaystyle\int_0^\infty{\cos{x}dx}$

##### SOLUTION
Let us first evaluate;

$\displaystyle I=\int{e^{\displaystyle-sx}\sin{x}dx}$ and $\displaystyle J=\int{e^{\displaystyle-sx}\cos{x}dx}$
Using integer by parts, we get
$I=-e^{\displaystyle-sx}\cos{x}-sJ$ (i)
$J=e^{\displaystyle-sx}\sin{x}+sI$ (ii)
Subtracting equations (i) and (ii), we get
$\displaystyle I=-e^{\displaystyle-sx}\left|\frac{\cos{x}+s\sin{x}}{1+s^2}\right|$
$\displaystyle\implies J=e^{\displaystyle-sx}\left|\sin{x}-\frac{s^2}{s^2+1}\sin{x}-\frac{s}{s^2+1}\cos{x}\right|$
$\displaystyle e^{\displaystyle-sx}\left|\frac{\sin{x}-s\cos{x}}{1+s^2}\right|$
Thus, $\displaystyle\int_0^\infty{e^{\displaystyle-sx}\sin{x}dx}=\frac{1}{s^2+1}$
$\displaystyle\int_0^\infty{e^{\displaystyle-sx}\cos{x}dx}=\frac{s}{s^2+1}$
Now, $\displaystyle\int_0^\infty{\cos{x}dx}=\lim_{s\rightarrow0}{\int_0^\infty{e^{\displaystyle-sx}\cos{x}dx}}=\lim_{s\rightarrow0}{\frac{s}{s^2+1}}=0$

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One Word Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
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#### Realted Questions

Q1 Single Correct Medium
$\int_{\pi /4}^{\pi /2} Cotx.dx_{=}$
• A. 2 log 2
• B. $\displaystyle \frac{\pi}{2}$ log2
• C. $\log 2$
• D. $\log\sqrt{2}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Evaluate the following integral:
$\displaystyle \int { \cfrac { \left( x+1 \right) { e }^{ x } }{ \sin ^{ 2 }{ \left( x{ e }^{ x } \right) } } } dx$

1 Verified Answer | Published on 17th 09, 2020

Q3 Assertion & Reason Hard
##### ASSERTION

$\displaystyle \int_{-1}^{1}\frac{\sin x-x^{4}}{4-\left | x \right |}dx$ is same as $\displaystyle \int_{0}^{1}\frac{-2x^{4}}{4-\left | x \right |}dx$

##### REASON

$\displaystyle \int_{-1}^{1}\left ( f\left ( x \right )+g\left ( x \right ) \right )dx=2\displaystyle \int_{0}^{1}f\left ( x \right )dx$ if $g(x)$ is an odd function and $f(x)$ is an even function.

• A. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• B. Assertion is correct but Reason is incorrect
• C. Both Assertion and Reason are incorrect
• D. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Integrate the function   $\displaystyle \frac {x}{e^{x^2}}$

1 Verified Answer | Published on 17th 09, 2020

Q5 Passage Hard
Let us consider the integral of the following forms
$f{(x_1,\sqrt{mx^2+nx+p})}^{\tfrac{1}{2}}$
Case I If $m>0$, then put $\sqrt{mx^2+nx+C}=u\pm x\sqrt{m}$
Case II If $p>0$, then put $\sqrt{mx^2+nx+C}=u\pm \sqrt{p}$
Case III If quadratic equation $mx^2+nx+p=0$ has real roots $\alpha$ and $\beta$, then put $\sqrt{mx^2+nx+p}=(x-\alpha)u\:or\:(x-\beta)u$