Mathematics

# $\int cos^{2}\theta d\theta$ = ?

##### SOLUTION
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$\begin{matrix} \int { { { \cos }^{ 2 } }\theta d\theta } \\ \Rightarrow \int { \dfrac { { \left( { 1-\cos 2\theta } \right) d\theta } }{ 2 } } \\ \Rightarrow \int { \dfrac { 1 }{ 2 } } d\theta -\dfrac { 1 }{ 2 } \cos 2\theta d\theta \\ \Rightarrow \dfrac { \theta }{ 2 } -\dfrac { 1 }{ 2 } \dfrac { { \sin 2\theta } }{ 1 } +C \\ \Rightarrow \dfrac { \theta }{ 2 } -\dfrac { { \sin 2\theta } }{ 4 } +C \\ \end{matrix}$

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

#### Realted Questions

Q1 Subjective Medium
Evaluate:
$\displaystyle \int _{ 1 }^{ e }{ \left( \frac{1}{\sqrt{x \ln x}} + \sqrt{\frac{\ln x}{x}} \right) dx }$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
The value of $\displaystyle \int_0^{\cfrac {\pi}{2}}\log \left (\frac {4+3 \sin x}{4+3 \cos x}\right )dx$ is
• A. $2$
• B. $\frac {3}{4}$
• C. $-2$
• D. $0$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
Evaluate: $\displaystyle \int \frac{\left ( \sin ^{-1} x\right )^{3}}{\sqrt{\left ( 1-x^{2} \right )}}dx$
• A. $\displaystyle \frac{1}{2}\left ( \sin ^{-1}x \right )^{4}$
• B. $\displaystyle\left ( \sin ^{-1}x \right )^{4}$
• C. $\displaystyle \frac{1}{3}\left ( \sin ^{-1}x \right )^{3}$
• D. $\displaystyle \frac{1}{4}\left ( \sin ^{-1}x \right )^{4}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
$\displaystyle \int\dfrac{\sec^{2}x}{(1+\tan x)^{3}}dx=$
• A. $\displaystyle \dfrac{1}{2(1+\tan x)^{3}}+c$
• B. $\displaystyle \dfrac{1}{(1+\tan x)^{3}}+c$
• C. $\displaystyle \dfrac{-1}{(1+\tan x)^{2}}+c$
• D. $\displaystyle \dfrac{-1}{2(1+\tan x)^{2}}+c$

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.

$\int u(x)\, v(x)dx\, =\, u(x)\, v_{1}(x)\, -\, u^{}(x)v_{2}(x)\, +\, u^{}(x)\, v_{3}(x)\, -\, .\, +\, (-1)^{n\, -\, 1}u^{n\, -\, 1}(x)v_{n}(x)\, -\, (-1)^{n\, -\, 1}$ $\int\, u^{n}(x)v_{n}(x)\, dx$ where $v_{1}(x)\, =\, \int v(x)dx,\, v_{2}(x)\, =\, \int v_{1}(x)\, dx\, ..\, v_{n}(x)\, =\, \int v_{n\, -\, 1}(x) dx$

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 $\int P_{n}(x)\, Q(x)\, dx$, where $P_{n}(x)$, is polynomial of degree n and the factor Q(x) is such that it can be integrated successively n + 1 times.