Q.469. Indefinite integration.

Question 469.
Integrate cosec^5 (5x) dx.

Answer 469.
Let 5x = u
=> 5 dx = du
=> Integrand
= (1/5) ∫ cosec^5 u du ... ( 1 )

∫ cosec^3 u du
= ∫ cosecu * cosec^2 u du
Using integration by parts
= cosecu ∫ cosec^2 u du - ∫ [d/du(cosecu) ∫ cosec^2u du] du
= - cosecu cotu - ∫ cosecu cot^2 u du
= - cosecu cotu - ∫ cosecu (cosec^2 u - 1) du
= - cosecu cotu - ∫ cosec^3 u du + ∫ cosecu du
=>
2 ∫ cosec^3 u du
= - cosecu cotu + ln ltan(u/2)l + 2c
=> ∫ cosec^3 u du
= - (1/2) cosecu cotu + (1/2) ln ltan(u/2)l + c ... ( 2 )

Now, ∫ cosec^5 u du
= ∫ cosec^3 u * cosec^2 u du
Integrating by parts,
= cosec^3 u ∫ cosec^2 u du - ∫ [d/du(cosec^3 u) ∫ cosecu du] du
= - cosec^3 u * cotu - ∫ 3 cosec^3 u * cot^2u du
= - cosec^3 u * cotu - 3 ∫ cosec^3u (cosec^2u - 1) du
= - cosec^3 u * cotu - 3 ∫ cosec^5 u du + 3 ∫ cosec^3 u du
=> 4 ∫ cosec^5 u du
= - cosec^3 u * cotu + 3 ∫ cosec^3 u du

[Plugging the value of ∫ cosec^3 u du from ( 2 ) above]
= - cosec^3 u * cotu + 3 [ - (1/2) cosecu cotu + (1/2) ln ltan(u/2)l ] + 4c
=> ∫ cosec^5 u du
= - (1/4) cosec^3 u * cotu - (3/8) cosecu cotu + (3/8) ln ltan(u/2)l ] + c

[Plugging in ( 1 ) above]
=> Integrand
= - (1/20) cosec^3 (5x) * cot(5x) - (3/40) cosec(5x) cot(5x)
   + (3/40) ln ltan(5x/2)l + c'. [c' = c/5]

Confirmation that the above answer is correct as verified by Wolfram Alpha:
Wolfram Alpha Link

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Q.468. Static Equilibrium, Application of Lami's Theorem

Question 468.
An object of mass 3kg is suspended by two light, inextensible strings. The strings make angles of 30degrees and 40degrees to the horizontal.
Find the magnitude of the tension in each spring.

Answer 468.

Refer to the figure as shown.

T = tension in the string making an angle of 30° to the horizontal

T ' = tension in the string making an angle of 40° to the horizontal

By Lami's theorem,
T/sin(90° + 40°) = T '/sin(90° + 30°) = 3 * 9.81/sin(180° - 40° - 30°)

=> T = (3 * 9.81) * cos40°/sin70° N = 24 N
and T ' = (3 * 9.81) * cos30°/sin70° N = 27.1 N.

For Lami's theorem, please refer to the following link:
Lami's Theorem

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