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This is a doubt from Integral calculus

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$\\1.\int \sqrt{\left ( \frac{\sin(x-a)}{\sin(x+a)} \right )}dx\\2.\int \frac{2\sin2x-\cos x}{6-\cos^2x-4\sin x}$

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H Himanshu Meshram

@Vinod

$\\\int \sqrt{\frac{\sin \left(x-a\right)}{\sin \left(x+a\right)}}dx\\rationalize\;it\\\int \sqrt{\frac{\sin \left(x-a\right)\sin \left(x-a\right)}{\sin \left(x+a\right)\sin \left(x-a\right)}}dx\\\int \frac{\sin \left(x-a\right)}{\sqrt{\sin \:\left(x+a\right)\sin \:\left(x-a\right)}}dx\\\because \sin \left(A+B\right)\sin \left(A-B\right)=\sin ^2A-\sin ^2B\\\int \frac{\sin x\:\cos a-\sin a\:\cos x}{\sqrt{\sin ^2x-\sin ^2a}}dx\\\cos a\int \frac{\sin x\:}{\sqrt{\sin ^2x-\sin ^2a}}dx\:-\sin a\int \:\frac{\:\cos \:x}{\sqrt{\sin \:^2x-\sin \:^2a}}dx\:\:\:$

$\\\cos a\int \frac{\sin x\:}{\sqrt{\sin ^2x-\sin ^2a}}dx\:\\\cos \:a\int \frac{\sin \:x\:}{\sqrt{1-\cos ^2x-1+\cos ^2a}}dx\:\\\Rightarrow \cos \:a\int \frac{\sin \:x\:}{\sqrt{\cos \:^2a-\cos ^2x}}dx\:\\put\;\cos x=t,and\;solve\\same\;method\;apply\;for\;\sin a\int \:\frac{\:\cos \:x}{\sqrt{\sin \:^2x-\sin \:^2a}}dx\:\:\:$

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Why is that once a person starts taking alcohol or drugs, it is difficult to get rid of this habit. Discuss it with your teacher.

16. Why is that once a person starts taking alcohol or drugs, it is difficult to get rid of this habit. Discuss it with your teacher.

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B Bipasha Kumar

It's called dependence and addiction. If the use or intake of alcohol is suddenly discontinued it leads to withdrawal symptoms.

Directions : Five friends P, Q, R, S, T, traveled to different cities Madras, Calcultta, Delhi, Bangalore, Hyderbad by different modes of transport by, bus, Train, Aeroplane, Car and Boat from Bombay.

(i) The person who travelled to Delhi did not travel by boat.

(ii) R went to Bangalore by car and Q went to Calcultta by aeroplane.

(iii) S travelled by boat whereas T travelled by train.

(iv) Bombay is not connected by bus to Delhi and Madras.

Which of the following combinations of the persons and mode is not correct?

Option 1)
T-Aeroplane
Option 2)
P-Bus
Option 3)
None of these
Option 4)
S-Boat
Option 5)
R-Car

Statements:

I. All papers are bags. II. No bag is green.

Conclusions:

I. No paper is green II. Some papers are green.

III. Some green are papers IV. Some bag are papers.

Option 1)
Either I or II follows
Option 2)
Either II or III follows
Option 3)
Only I and III follow
Option 4)
Only I and IV follow
Option 5)
None of these

Statements:

I. All books are cakes II. All cakes are apples.

Conclusions:

I. Some cakes are books. II. No cake is book.

III. Some apples are books. IV. All apples are books.

Option 1)
Only I follow
Option 2)
Only either I or II follows
Option 3)
Only I and III follow
Option 4)
Only either III or IV follows
Option 5)
None follows

Directions : Given two statements, verify the conclusions and mark the answer as given below.

Statements:

I. Some chairs are stools. II. Table is a chair.

Conclusions:

I. Some stools are chairs II. Table is not a stool.

Option 1)
if only conclusion I follow.
Option 2)
If only conclusion II follows.
Option 3)
If either conclusion I or II follows.
Option 4)
If neither of the two conclusions follows.
Option 5)
If both conclusions follow

Statements:

I. Some Phone are watches. II. All watches are guns.

Conclusions:

I. All guns are watches. II. Some guns are phones.

Option 1)
if only conclusion I follow.
Option 2)
If only conclusion II follows.
Option 3)
If either conclusion I or II follows.
Option 4)
If neither of the two conclusions follows.
Option 5)
If both conclusions follow

Directions : Given two statements, verify the conclusions and mark the answer as given below.

Statements:

I. Some stones are cups. II. Some cups are black.

Conclusions:

I. Some black are not cups II. Some cups are stones

Option 1)
if only conclusion I follow.
Option 2)
If only conclusion II follows.
Option 3)
If either conclusion I or II follows.
Option 4)
If neither of the two conclusions follows.
Option 5)
If both conclusions follow

Directions : Given two statements, verify the conclusions and mark the answer as given below.

Statements:

I. Some Calendars are sticks. II. No stick is flower.

Conclusions:

I. Some Calendars are flowers. II. No calendar is flower

Option 1)
if only conclusion I follow.
Option 2)
If only conclusion II follows.
Option 3)
If either conclusion I or II follows.
Option 4)
If neither of the two conclusions follows.
Option 5)
If both conclusions follow

Directions : Given two statements, verify the conclusions and mark the answer as given below.

Statements:

I. Some kites are horses. II. All Horses are dogs.

Conclusions:

I. All dogs are horses. II. Some dogs are horses.

Option 1)
if only conclusion I follow.
Option 2)
If only conclusion II follows.
Option 3)
If either conclusion I or II follows.
Option 4)
If neither of the two conclusions follows.
Option 5)
If both conclusions follow

Statements:

I. Some barbers are painters II. No painter are watches

Conclusions:

I. Some barbers are not watches II. Some barbers are watches

III. Some watches are not barbers IV. Some watches are barbers

Option 1)
1 & 3 follow
Option 2)
Either 1 or 2 & 3 follow
Option 3)
None of These
Option 4)
Either 3 or 4 & 1 follow
Option 5)
Only 1 follow

Statements:

I. All classes are books II. All pens are books

Conclusions:

I Some classes are pens II. Some pens are books

III. Some pens are classes IV. Some classes are books

Option 1)
None of These
Option 2)
Only 2 follow
Option 3)
1 & 2 follow
Option 4)
2 & 4 follow
Option 5)
Only 1 follow

Statements:

I. Some boxes are pens II. All dogs are pens

Conclusions:

I. Some boxes are pens II. Some pens are boxes

III. Some pens are dogs IV. All pens are dogs

Option 1)
1,2 & 3 follow
Option 2)
2,3 & 4 follows
Option 3)
None of These
Option 4)
1,2 & 4 follows
Option 5)
1,3 & 4 follows

Directions : Given two statements, verify the conclusions and mark the answer as given below.

Statements:

I. All plays are stories II. Some poems are plays

Conclusions:

I. Some poems are stories II. All stories are poems

Option 1)
if only conclusion I follow.
Option 2)
If only conclusion II follows.
Option 3)
If either conclusion I or II follows.
Option 4)
If neither of the two conclusions follows.
Option 5)
If both conclusions follow

Directions : Given two statements, verify the conclusions and mark the answer as given below.

Statements:

I. All women are mothers II. All mothers are sisters

Conclusions:

I. All womens are sisters II. Some women are not sisters

Option 1)
if only conclusion I follow.
Option 2)
If only conclusion II follows.
Option 3)
If either conclusion I or II follows.
Option 4)
If neither of the two conclusions follows.
Option 5)
If both conclusions follow

Directions : Given two statements, verify the conclusions and mark the answer as given below.

Statements:

I. Some boys are thieves II. All thieves are dacoits

Conclusions:

I. Some boys are dacoits II. All dacoits are thieves

Option 1)
if only conclusion I follow.
Option 2)
If only conclusion II follows.
Option 3)
If either conclusion I or II follows.
Option 4)
If neither of the two conclusions follows.
Option 5)
If both conclusions follow

Directions : Given two statements, verify the conclusions and mark the answer as given below.

Statements:

I. All grasses are trees II. No tree is a shrub

Conclusions:

I. No grasses are shrubs II. Some shrubs are glasses

Option 1)
if only conclusion I follow.
Option 2)
If only conclusion II follows.
Option 3)
If either conclusion I or II follows.
Option 4)
If neither of the two conclusions follows.
Option 5)
If both conclusions follow

Directions : Given two statements, verify the conclusions and mark the answer as given below.

Statements:

I. All fans are chairs II. No tables are fans

Conclusions:

I. No tables are chairs II. Some tables are chairs

Option 1)
if only conclusion I follow.
Option 2)
If only conclusion II follows.
Option 3)
If either conclusion I or II follows.
Option 4)
If neither of the two conclusions follows.
Option 5)
If both conclusions follow

Directions : Given two statements, verify the conclusions and mark the answer as given below.

Statements:

I. Some bottles are pencils II. Some pencils are glasses

Conclusions:

I. No glass is bottle II. Some bottles are glasses

Option 1)
if only conclusion I follow.
Option 2)
If only conclusion II follows.
Option 3)
If either conclusion I or II follows.
Option 4)
If neither of the two conclusions follows.
Option 5)
If both conclusions follow

Directions : Given two statements, verify the conclusions and mark the answer as given below.

Statements:

I. Some dogs are bulls II. No tigers are dogs

Conclusions:

I. No dogs are tiger II. Some bulls are tiger

Option 1)
if only conclusion I follow.
Option 2)
If only conclusion II follows.
Option 3)
If either conclusion I or II follows.
Option 4)
If neither of the two conclusions follows.
Option 5)
If both conclusions follow

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