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What is the antiderivative of #sinx^3#?

Answer 1

Emily Ammerman

If you mean #(sinx)^3#, please see below. If you mean #sin(x^3)#, I can't help.

Here is one possibility

#int (sinx)^3 dx = int sin^2x sinx dx#

# = int ((1-cos^2 x) sinx dx #

# = int sinx dx + int underbrace(cos^2x)_u underbrace((-sinx) dx)_(du)#

# = -cosx + cos^3x/3 +C#

As is typical of expressions involving trigonometric functions, there are other ways to express this family of functions.

(WolframAlpha gives #1/12(cos(3x)-9cosx)+C# and simplifies the difference between our basic answers to #0#.)

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Answer 2

Emilia Aguilar

The antiderivative of sin(x^3) with respect to x cannot be expressed in terms of elementary functions. However, it can be approximated numerically or represented using special functions.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.