Mathematics
One of the roots of the equation $$2000x^6+100x^5+10x^3+x-2=0$$ is of the form $$\dfrac{m+\sqrt{n}}{r}$$. When 'm' is non zero integer and n and r relatively prime natural numbers. Then $$\dfrac{m+n+r}{100}=?$$
Single Correct
Medium
Published on 17th 09, 2020
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Q1
Single Correct
Hard
$$\displaystyle \int { \cfrac { { \sin }^{ 6 }x+{ \cos }^{ 6 }x }{ { \sin }^{ 2 }x{ \cos }^{ 2 }x } } dx$$
- A. $$\sin x-\cot x+C$$
- B. $$\cos x-\cot x+C$$
- C. All of the above
- D. $$\tan x-\cot x-3x+C$$
Asked in: Mathematics - Integrals
1 Verified Answer | Published on 17th 09, 2020
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Q2
Subjective
Medium
Solve $$\displaystyle \int {\frac{x}{{1 - {x^3}}}dx} $$
Asked in: Mathematics - Integrals
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Q3
Subjective
Medium
Evaluate the integral $$\displaystyle \int_0^{\tfrac {\pi}{2}}\frac {\sin x}{1+\cos^2x}dx$$ using substitution.
Asked in: Mathematics - Integrals
1 Verified Answer | Published on 17th 09, 2020
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Q4
Subjective
Medium
Evaluate the following integral:
$$\displaystyle \int { co\sec { x } \log { \left( co\sec { x } -\cot { x } \right) } } dx\quad $$
$$\displaystyle \int { co\sec { x } \log { \left( co\sec { x } -\cot { x } \right) } } dx\quad $$
Asked in: Mathematics - Integrals
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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
$$\displaystyle \int u\left ( x \right )v\left ( x \right )dx=u\left ( x \right )v_{1}-u'\left ( x \right )v_{2}\left ( x \right )+u''\left ( x \right )v_{3}\left ( x \right )+...+\left ( -1 \right )^{n-1}u^{n-1}\left ( x \right )V_{n}\left ( x \right ) \\ -\left ( -1 \right )^{n-1}\int u^{n}\left ( x \right )V_{n}\left ( x \right )dx$$
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.
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
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