To determine

To find:

∫1-sin3⁡tsin2⁡t dt

Answer

-cot⁡t+cos⁡t+C

Explanation

1)  Concept:

Apply the formula of indefinite integral and simplify.

2) Formula:

∫cosec2⁡θ dθ= -cot⁡θ+C

∫sin⁡θ dθ= -cos⁡θ+C

1sin2⁡θ=cosec2θ

3) Calculation:

Consider,

∫1-sin3⁡tsin2⁡t dt

After separating the denominator,

∫1-sin3⁡tsin2⁡t dt=∫(1sin2⁡t dt-sin⁡t dt)

The integral can be rewritten as

= ∫cosec2t dt-∫sin⁡t dt

By applying the formula of integral, we get

=-cot⁡t-(-cos⁡t)+C

=-cot⁡t+cos⁡t+C

Conclusion:

∫1-sin3⁡tsin2⁡t dt= -cot⁡t+cos⁡t+C