Mathematics

# Integrate:$\displaystyle \int_{0}^{\pi }{\left( \sin mx\cdot \cos nx \right)}dx$, where $(m \ne n, \in to\,\, integer)$

$0$

##### SOLUTION

Consider the given integral.

$I=\int_{0}^{\pi }{\left( \sin mx\cdot \cos nx \right)}dx$

We know that

$\sin A\cdot \sin B=\dfrac{1}{2}\left[ \cos \left( A-B \right)-\cos \left( A+B \right) \right]$

Therefore,

$I=\int_{0}^{\pi }{\left( \dfrac{1}{2}\left( \cos \left( mx-nx \right)-\cos \left( mx+nx \right) \right) \right)}dx$

$I=\dfrac{1}{2}\int_{0}^{\pi }{\left( \cos \left( m-n \right)x-\cos \left( m+n \right)x \right)}dx$

$I=\dfrac{1}{2}\int_{0}^{\pi }{\left( \cos \left( m-n \right)x \right)}dx-\dfrac{1}{2}\int_{0}^{\pi }{\left( \cos \left( m+n \right)x \right)}dx$

$I=\dfrac{1}{2}\left[ \dfrac{\sin \left( m-n \right)x}{m-n} \right]_{0}^{\pi }-\dfrac{1}{2}\left[ \dfrac{\sin \left( m+n \right)x}{m+n} \right]_{0}^{\pi }$

$I=\dfrac{1}{2}\left[ \dfrac{\sin \left( m-n \right)\pi }{m-n}-\dfrac{\sin \left( m-n \right)0}{m-n} \right]-\dfrac{1}{2}\left[ \dfrac{\sin \left( m+n \right)\pi }{m+n}-\dfrac{\sin \left( m+n \right)0}{m+n} \right]$

$I=0$

Its FREE, you're just one step away

Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Single Correct Medium
$\displaystyle \int \frac {ln \left ( x \right )} {x\sqrt{1 + ln \left ( x \right )}} dx$ equlas
• A. $\displaystyle \frac{4}{3} \sqrt{1 + ln \left | x \right |} (ln \left | x \right | - 2) - c$
• B. $\displaystyle \frac{1}{3} \sqrt{1 + ln \left | x \right |} (ln \left | x \right | - 2) + c$
• C. $\displaystyle 2 \sqrt{1 + ln \left | x \right |} (3ln \left | x \right | - 2) + c$
• D. $\displaystyle \frac{2}{3} \sqrt{1 + ln \left | x \right |} (ln \left | x \right | - 2) + c$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
Let $A=\displaystyle \int_0^1 \dfrac{e^t }{1+t}dt$ then $\displaystyle \int_{a-1}^a\dfrac{e^{-t}}{t-a-1}fdt$ has the value
• A. $-Ae^{-a}$
• B. $-Ae^{a}$
• C. None of these
• D. $Ae^{-a}$

1 Verified Answer | Published on 17th 09, 2020

Q3 One Word Medium
Evalute $\displaystyle \int_{0}^{\pi /2}\frac{\sin ^{2}x-\cos ^{2}x}{\sin ^{3}x+\cos ^{3}x}dx =$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
Evaluate the integral
$\displaystyle \int_{0}^{\pi /4}\log(1+\tan x)dx$
• A. $\pi log2$
• B. $\displaystyle \frac{\pi}{4} log2$
• C. $-\pi log2$
• D. $\displaystyle \frac{\pi}{8} log2$

Consider two differentiable functions $f(x), g(x)$ satisfying $\displaystyle 6\int f(x)g(x)dx=x^{6}+3x^{4}+3x^{2}+c$ & $\displaystyle 2 \int \frac {g(x)dx}{f(x)}=x^{2}+c$. where $\displaystyle f(x)>0 \forall x \in R$