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Chapter 5: Q. 50 (page 452)

Solve each of the integrals in Exercises 21–66. Some of the integrals require the methods presented in this section, and some do not. (The last four exercises involve hyperbolic functions.)

∫ csc 3 x cot3xdx

Short Answer

Expert verified

The solution of the integral is-csc5x5+csc3x3.

Step by step solution

01

Step 1. Given Information

The given integral is∫ csc 3 x cot3xdx.

02

Step 2. Rewrite and substitute

  • Use the trigonometric identities to rewrite the given integral.

∫ csc 3 x cot3xdx=∫ csc 2 x  csc 2 x -1 csc  x cotxdx

  • If cscx=u,-cscxcotxdx=du.
  • Substitute the values into the integral and simplify.

∫ csc 2 x  csc 2 x -1 csc  x cotxdx=∫ -u2u2-1du=-∫ u4-u2du=-∫ u4du-∫ u2du

03

Step 3. Integrate

  • Perform integration on the obtained integral.

-∫ u4du-∫ u2du=-u55-u33

  • Substitute cscx=uto obtain the solution of the given integral.

-u55-u33=-csc5x5-csc3x3=-csc5x5+csc3x3

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