Mathematics

What is the value of $$\int_{0}^{\pi}\dfrac {dx}{5-4\cos x}$$?


ANSWER

$$\dfrac {\pi \log 2}{32}$$


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Single Correct Medium Published on 17th 09, 2020
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Q1 Single Correct Medium
The value of $$\displaystyle \int { { e }^{ x }\frac { 1+n{ x }^{ n-1 }-{ x }^{ 2n } }{ \left( 1-{ x }^{ n } \right) \sqrt { 1-{ x }^{ 2n } }  } dx } $$ is 
  • A. $$\displaystyle { e }^{ x }\frac { \sqrt { 1-{ x }^{ n } } }{ 1-{ x }^{ n } } +c$$
  • B. $$\displaystyle { e }^{ x }\frac { \sqrt { 1+{ x }^{ 2n } } }{ 1-{ x }^{ 2n } } +c$$
  • C. $$\displaystyle { e }^{ x }\frac { \sqrt { 1-{ x }^{ 2n } } }{ 1-{ x }^{ 2n } } +c$$
  • D. $$\displaystyle { e }^{ x }\frac { \sqrt { 1-{ x }^{ 2n } } }{ 1-{ x }^{ n } } +c$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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Q2 Single Correct Hard
The value of $$\displaystyle \int_{-2}^{0}\left \{ x^{3}+3x^{2}+3x+3+(x+1)cos(x+1) \right \}dx$$ is
  • A. $$-4$$
  • B. $$0$$
  • C. $$6$$
  • D. $$4$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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Q4 Single Correct Medium
$$\displaystyle\int {\dfrac{{{\text{ln}}\left( {{\text{e}}{{\text{x}}^{\text{x}}}} \right)}}{{{\text{x}}{{\text{e}}^{\text{x}}}{{\left( {{\text{lnx}}} \right)}^{\text{2}}}}}} {\text{dx}}\;{\text{is}}$$
  • A. $$\dfrac{{\text{1}}}{{{{\text{e}}^{\text{x}}}{\text{lnx}}}}{\text{ + c}}$$
  • B. $$\dfrac{{{e^2}}}{{{{\text{e}}^{\text{x}}}{\text{lnx}}}}{\text{ + c}}$$
  • C. $$ - \dfrac{1}{{{{\text{e}}^{{\text{ - x}}}}{\text{lnx}}}}{\text{ + c}}$$
  • D. $$ - \dfrac{1}{{{{\text{e}}^{\text{x}}}{\text{lnx}}}}{\text{ + c}}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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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 :

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1 Verified Answer | Published on 17th 08, 2020

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