`int (1 - csc(t)cot(t)) dt` Find the indefinite integral.

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Chapter 4, 4.1 - Problem 27 - Calculus of a Single Variable (10th Edition, Ron Larson).
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gsarora17 | (Level 2) Associate Educator

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`int(1-csc(c)cot(t))dt`

apply the sum rule,

`=int1dt-intcsc(t)cot(t)dt`

`=t-int((1/sin(t)(cos(t)/sin(t)))dt`

`=t-intcos(t)/(sin^2(t))dt`

Now, let sin(t)=x `rArr` cos(t)dt=dx

`=t-intdx/x^2`

apply the power rule

`=t-(x^(-2+1)/(-2+1))` 

`=t-(x^-1/(-1))`

`=t+1/x`

plug back x=sin(t) and add constant C to the solution,

`=t+1/sin(t)+C`

 

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