Mathematics

# Evaluate the following integral$\int { \cfrac { \cos { 2x } }{ { \left( \cos { x } +\sin { x } \right) }^{ 2 } } } dx\quad$

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

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

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

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

Let $t=\cos{x}+\sin{x}\Rightarrow\,dt=\left(-\sin{x}+\cos{x}\right)dx$

$I=\displaystyle\int{\dfrac{dt}{t}}$

$I=\log{\left|t\right|}+c$

$\therefore\,I=\log{\left|\cos{x}+\sin{x}\right|}+c$ where $t=\cos{x}+\sin{x}$

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 Subjective Medium
$\displaystyle \int{ \sin }^{ 4 }x \cos x dx$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Find $\int{\dfrac{dx}{x^2-a^2}}$ and hence evaluate $\int{\dfrac{dx}{3x^2+13x-10}}$.

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Integrate  $\int x.sin2xdx$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
Evaluate: $\displaystyle \int \frac{5x^{8}+7x^{6}}{\left ( x^{2}+1+2x^{7} \right )^{2}}dx$
• A. $\displaystyle \frac{2x^{7}}{2x^{7}+x^{2}+1}$
• B. $\displaystyle \frac{x^{6}}{2x^{7}+x^{2}+1}$
• C. $\displaystyle \frac{x^{14}}{2x^{7}+x^{2}+1}$
• D. $\displaystyle \frac{x^{7}}{2x^{7}+x^{2}+1}$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q5 Passage Hard
In calculating a number of integrals we had to use the method of integration by parts several times in succession.
The result could be obtained more rapidly and in a more concise form by using the so-called generalized formula for integration by parts
$\displaystyle \int u\left ( x \right )v\left ( x \right )dx=u\left ( x \right )v_{1}-u'\left ( x \right )v_{2}\left ( x \right )+u''\left ( x \right )v_{3}\left ( x \right )+...+\left ( -1 \right )^{n-1}u^{n-1}\left ( x \right )V_{n}\left ( x \right ) \\ -\left ( -1 \right )^{n-1}\int u^{n}\left ( x \right )V_{n}\left ( x \right )dx$
where  $\displaystyle v_{1}\left ( x \right )=\int v\left ( x \right )dx,v_{2}\left ( x \right )=\int v_{1}\left ( x \right )dx ..., v_{n}\left ( x \right )= \int v_{n-1}\left ( x \right )dx$
Of course, we assume that all derivatives and integrals appearing in this formula exist. The use of the generalized formula for integration by parts is especially useful when  calculating $\displaystyle \int P_{n}\left ( x \right )Q\left ( x \right )dx,$ where $\displaystyle P_{n}\left ( x \right )$ is polynomial of degree n and the factor Q(x) is such that it can be integrated successively n+1 times.

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020