Mathematics

# $\int ^{\pi /2}_{-\pi /2} cost . \sin(2t-\displaystyle \frac{\pi}{4})dt=$

$-\displaystyle \frac{\sqrt{2}}{3}$

##### SOLUTION
$I=\int { \left( cost \right) \left( sin\left( 2t-\cfrac { \pi }{ 4 } \right) \right) } dt=-\int { sin\left( \cfrac { \pi }{ 4 } -2t \right) \left( cost \right) dt } \\ =-\cfrac { 1 }{ 2 } \int { \left( sin\left( \cfrac { 1 }{ 4 } \left( \pi -12t \right) \right) +sin\left( \cfrac { 1 }{ 4 } \left( \pi -4t \right) \right) \right) dt } \\ =-\cfrac { 1 }{ 2 } \int { sin\left( \cfrac { 1 }{ 4 } \left( \pi -12t \right) \right) dt } -\cfrac { 1 }{ 2 } \int { sin\left( \cfrac { 1 }{ 4 } \left( \pi -4t \right) \right) dt }$
Let ${ I }={ I }_{ 1 }+{ I }_{ 2 }$, such that
${ I }_{ 1 }=-\cfrac { 1 }{ 2 } \int { sin\left( \cfrac { 1 }{ 4 } \left( \pi -12t \right) \right) dt }$
Substituting $x=\pi -4t\Rightarrow dx=-4dt$
${ I }_{ 1 }=\cfrac { 1 }{ 8 } \int { sin\cfrac { x }{ 4 } dx } =-\cfrac { 1 }{ 2 } cos\cfrac { x }{ 4 } =-\cfrac { 1 }{ 2 } sin\left( 3t+\cfrac { \pi }{ 4 } \right)$ ...(1)
And
${ I }_{ 2 }=-\cfrac { 1 }{ 2 } \int { sin\left( \cfrac { 1 }{ 4 } \left( \pi -4t \right) \right) dt }$
Substituting $y=\pi -12t\Rightarrow dy=-12dt$, we get
${ I }_{ 2 }=\cfrac { 1 }{ 24 } \int { sin\cfrac { y }{ 4 } dy } =-\cfrac { 1 }{ 6 } cos\cfrac { y }{ 4 } \quad =-\cfrac { 1 }{ 6 } sin\left( t+\cfrac { \pi }{ 4 } \right)$...(2)
Therefore from (1) and (2)
$\int _{ -\cfrac { \pi }{ 2 } }^{ \cfrac { \pi }{ 2 } }{ \left( cott \right) sin\left( 2t-\cfrac { \pi }{ 4 } \right) dt } =\left[ -\cfrac { 1 }{ 2 } sin\left( 3t+\cfrac { \pi }{ 4 } \right) -\cfrac { 1 }{ 6 } sin\left( t+\cfrac { \pi }{ 4 } \right) \right] _{ -\cfrac { \pi }{ 2 } }^{ \cfrac { \pi }{ 2 } }=-\cfrac { \sqrt { 2 } }{ 3 }$

Its FREE, you're just one step away

Single Correct Hard Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 105

#### Realted Questions

Q1 Single Correct Hard
The value of $\displaystyle \int e^{x} \left [\dfrac {1 + \sin x}{1 + \cos x}\right ] dx$ is
• A. $\dfrac {1}{2} e^{x} \sec \dfrac {x}{2} + C$
• B. $e^{x} \sec \dfrac {x}{2} + C$
• C. $\dfrac {1}{2} e^{x} \tan \dfrac {x}{2} + C$
• D. $e^{x} \tan \dfrac {x}{2} + C$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Evaluate $\displaystyle \int \sin{mx}\sin{nx}\ dx$ on $R,\ m\neq n,\ m$ and $n$ are positive integers.

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Find an anti derivative (or integral) of the given function by the method of inspection.
$\sin 2x-4e^{3x}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Integrate the following w.r.t. $x$
$e^{2x} + \dfrac {1}{x^{2}}$.

Let $n \space\epsilon \space N$ & the A.M., G.M., H.M. & the root mean square of $n$ numbers $2n+1, 2n+2, ...,$ up to $n^{th}$ number are $A_{n}$, $G_{n}$, $H_{n}$ and $R_{n}$ respectively.