Mathematics

# Solve $\displaystyle \int _{\pi/6}^{\pi/3}\cos^{2}x dx$

##### SOLUTION

$\begin{array}{l} \int _{ \pi /6 }^{ \pi /3 }{ { { \cos }^{ 2 } }xdx } \\ =\dfrac { 1 }{ 2 } \int _{ \pi /6 }^{ \pi /3 }{ { { \cos }^{ 2 } }xdx } \\ =\dfrac { 1 }{ 2 } \int _{ \pi /6 }^{ \pi /3 }{ \left( { 1+\cos 2x } \right) dx } \\ =\dfrac { 1 }{ 2 } { \left[ { x+\dfrac { { \sin 2x } }{ 2 } } \right] _{ \pi /6 } }^{ \pi /3 } \\ =\dfrac { 1 }{ 2 } \left[ { \left( { \dfrac { \pi }{ 3 } +\dfrac { 1 }{ 2 } \sin \dfrac { { 2\pi } }{ 3 } } \right) -\left( { \dfrac { \pi }{ 6 } +\dfrac { 1 }{ 2 } +\sin \dfrac { \pi }{ 3 } } \right) } \right] \\ =\dfrac { 1 }{ 2 } \left[ { \left( { \dfrac { \pi }{ 3 } +\dfrac { 1 }{ 2 } \times \sin \dfrac { \pi }{ 3 } } \right) -\left( { \dfrac { \pi }{ 2 } +\dfrac { 1 }{ 2 } \times \dfrac { { \sqrt { 3 } } }{ 2 } } \right) } \right] \\ =\dfrac { 1 }{ 2 } \left[ { \left( { \dfrac { \pi }{ 3 } +\dfrac { 1 }{ 2 } \times \dfrac { { \sqrt { 3 } } }{ 3 } } \right) -\left( { \dfrac { \pi }{ 2 } +\dfrac { { \sqrt { 3 } } }{ 4 } } \right) } \right] \\ =\dfrac { 1 }{ 2 } \left[ { \dfrac { \pi }{ 3 } +\dfrac { { \sqrt { 3 } } }{ 4 } -\dfrac { \pi }{ 2 } -\dfrac { { \sqrt { 3 } } }{ 4 } } \right] \\ =\dfrac { 1 }{ 2 } \left[ { \dfrac { { -\pi } }{ 6 } } \right] \\ =\dfrac { { -\pi } }{ { 12 } } \end{array}$

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

#### Realted Questions

Q1 Single Correct Hard
Resolve $\displaystyle \frac{(x-a)(x-b)(x-c)}{(x-\alpha)(x-\beta)(x-\gamma)}$ into partial fractions.
• A. $\displaystyle 2+ \frac{(\alpha-a)(\alpha-b)(\alpha-c)}{(\alpha-\beta)(\alpha-\gamma)}\cdot \frac{1}{(x-\alpha)}+ \frac{(\beta-a)(\beta-b)(\beta-c)}{(\beta-\alpha)(\beta-\gamma)}\cdot \frac{1}{(x-\beta)}$
$\displaystyle+ \frac{(\gamma-a)(\gamma-b)(\gamma-c)}{(\gamma-\alpha)(\gamma-\beta)}\cdot \frac{1}{(x-\gamma)}$
• B. $\displaystyle \frac{(\alpha-a)(\alpha-b)(\alpha-c)}{(\alpha-\beta)(\alpha-\gamma)}\cdot \frac{1}{(x-\alpha)}+ \frac{(\beta-a)(\beta-b)(\beta-c)}{(\beta-\alpha)(\beta-\gamma)}\cdot \frac{1}{(x-\beta)}$
$\displaystyle+ \frac{(\gamma-a)(\gamma-b)(\gamma-c)}{(\gamma-\alpha)(\gamma-\beta)}\cdot \frac{1}{(x-\gamma)}$
• C. $\displaystyle 1- \frac{(\alpha-a)(\alpha-b)(\alpha-c)}{(\alpha-\beta)(\alpha-\gamma)}\cdot \frac{1}{(x-\alpha)}+ \frac{(\beta-a)(\beta-b)(\beta-c)}{(\beta-\alpha)(\beta-\gamma)}\cdot \frac{1}{(x-\beta)}$
$\displaystyle+ \frac{(\gamma-a)(\gamma-b)(\gamma-c)}{(\gamma-\alpha)(\gamma-\beta)}\cdot \frac{1}{(x-\gamma)}$
• D. $\displaystyle 1+ \frac{(\alpha-a)(\alpha-b)(\alpha-c)}{(\alpha-\beta)(\alpha-\gamma)}\cdot \frac{1}{(x-\alpha)}+ \frac{(\beta-a)(\beta-b)(\beta-c)}{(\beta-\alpha)(\beta-\gamma)}\cdot \frac{1}{(x-\beta)}$
$\displaystyle+ \frac{(\gamma-a)(\gamma-b)(\gamma-c)}{(\gamma-\alpha)(\gamma-\beta)}\cdot \frac{1}{(x-\gamma)}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\int_{-1}^{1}(\frac{x+x^{3}}{1+x^{4}})dx$ is equal to
• A. $2\pi$
• B. $\frac{\pi }{2}$
• C. $\pi$
• D.

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Evaluate the following integral:
$\int { \cfrac { 1+\sin { x } }{ \sqrt { x-\cos { x } } } } dx$

1 Verified Answer | Published on 17th 09, 2020

Q4 Assertion & Reason Hard
##### ASSERTION

STATEMENT-1 : $\displaystyle \int \frac{\left \{ f(x)\phi '(x)-f'(x)\phi (x) \right \}}{f(x)\phi (x)}\left \{ \log \phi (x)-\log f(x) \right \}dx=\frac{1}{2}\left \{ \log \frac{\phi(x)}{f(x)} \right \}^{2}+c$

##### REASON

STATEMENT-2 : $\displaystyle \int (h(x))^{n}h'(x)dx=\frac{(h(x))^{n+1}}{n+1}+c$

• A. STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1
• B. STATEMENT-1 is True, STATEMENT-2 is False
• C. STATEMENT-1 is False, STATEMENT-2 is True
• D. STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1

1 Verified Answer | Published on 17th 09, 2020

Q5 Passage Medium
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 $\displaystyle f(x)>0 \forall x \in R$

On the basis of above information, answer the following questions :

Asked in: Mathematics - Limits and Derivatives

1 Verified Answer | Published on 17th 08, 2020