Mathematics

# Solve $\int \dfrac{{\cos 2x}}{{{{\left( {\cos x + \sin x} \right)}^2}}} \ dx$

##### SOLUTION

We have,

$I=\int{\dfrac{\cos 2x}{{{\left( \cos x+\sin x \right)}^{2}}}}dx$

$I=\int{\dfrac{\cos 2x}{{{\cos }^{2}}x+{{\sin }^{2}}x+2\sin x\cos x}}dx$

$I=\int{\dfrac{\cos 2x}{1+\sin 2x}}dx$

Let $t=1+\sin 2x$

$\dfrac{dt}{dx}=0+2\cos 2x$

$\dfrac{dt}{2}=\cos 2xdx$

Therefore,

$I=\dfrac{1}{2}\int{\dfrac{dt}{t}}$

$I=\dfrac{1}{2}\ln \left( t \right)+C$

Put the value of $t$, we get

$I=\dfrac{1}{2}\ln \left( 1+\sin 2x \right)+C$

Its FREE, you're just one step away

Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Single Correct Medium
$\displaystyle \int\frac{\cos x}{(1+\sin x)(2+\sin x)}dx=$
• A. $\displaystyle \log|\frac{2+\sin x}{1+\sin x}|+c$
• B. $\log|(1+sinx)(2+\sin x)|+c$
• C. $\log|(1+sinx)+(2+\sin x)|+c$
• D. $\displaystyle \log|\frac{1+\sin x}{2+\sin x}|+c$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
If $\displaystyle \frac{1}{(x-2)(x-3)^{3}}=\frac{A}{x-2}+\frac{B}{x-3}+\frac{C}{(x-3)^{2}}+\frac{D}{(x-3)^{3}}$ then $B=$
• A. $-1$
• B. $\displaystyle \frac{1}{25}$
• C. $0$
• D. $1$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
$\displaystyle \int \frac{1}{2x^{2}+x-1}dx$

• A. $\displaystyle =\frac{1}{6}log\left | \frac{x-1/2}{x+1} \right |+C=\frac{1}{2}log\left | \frac{2x-1}{2\left ( x+1 \right )} \right |+C$
• B. $\displaystyle =\frac{1}{6}log\left | \frac{x+1/2}{x+1} \right |+C=\frac{1}{2}log\left | \frac{2x+1}{2\left ( x+1 \right )} \right |+C$
• C. None of these
• D. $\displaystyle =\frac{1}{3}log\left | \frac{x-1/2}{x+1} \right |+C=\frac{1}{3}log\left | \frac{2x-1}{2\left ( x+1 \right )} \right |+C$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
$\displaystyle \mathrm{I}\mathrm{f}\frac{\mathrm{x}^{2}}{(\mathrm{x}^{2}+1)(\mathrm{x}^{2}+2)}=\frac{\mathrm{A}\mathrm{x}+\mathrm{B}}{\mathrm{x}^{2}+1}+\frac{\mathrm{C}\mathrm{x}+\mathrm{D}}{\mathrm{x}^{2}+2}$ then $(A,C)=$
• A. $(1,-1)$
• B. $(1,1)$
• C. $(1,2)$
• D. $(0,0)$

Let $F: R\rightarrow R$ be a thrice differential function. Suppose that $F(1) = 0, F(3) = -4$ and $F'(x)<0$ for all $x\in\left(\dfrac{1}{2},3\right)$. Let $f(x) = xF(x)$ for all $x\in R$.