Finding the Antiderivative of sin^4(x) in Basic Calculus

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In summary, to find the antiderivative of (sin(x))^4, you can use the substitution method and reduce the power of sin4x to 2sin2x. From there, you can use the trigonometric identities to simplify the expression and find the antiderivative.
  • #1
calculateme
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I am taking basic calculus, and have just got to integration. Can someone please tell me how to find the antiderivative of (sin(x))^4?
 
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  • #2
Looks like home work. Replace a power with a multiple - you can do sin^2 using cos 2, so this is no harder.
 
  • #3
Sorry, but I still don't understand. How do you find the antiderivative of (sin(x))^2? Could you explain it to me please?
 
  • #4
Do you know about the chain rule?
do you know the derivatives of [tex]\sin x[/tex] and [tex]x^2[/tex]?
 
  • #5
Originally posted by NateTG
Do you know about the chain rule?
do you know the derivatives of [tex]\sin x[/tex] and [tex]x^2[/tex]?

Yes, but how are they going to help me find the antiderivative of [tex]sin^4x[/tex]?
 
  • #6
Try to Reduce the power of sin4x by
2sin2x=1-cos2x.

therefore
4sin4x=(1-cos2x)2

i.e 1+cos22x-2cos2x
Again use

2cos22x = 1+cos24x

Simplifying u will obtain

[tex]\sin^4x = \frac{3}{8} +\frac{cos4x}{8}-\frac{cos2x}{2}[/tex]

Hope this will help u
 
  • #7
Himanshu

I think you have a typo

2cos22x = 1+cos24x

Should be

2cos22x = 1+cos4x
 
  • #8
Ya typo is there it is

2cos22x = 1+cos4x

error is regretted
 
  • #9
Himanshu

Thanks a lot, I understand now.
 

What is a derivative?

A derivative is a mathematical concept that represents the rate of change of a function at a given point. It can also be thought of as the slope of a tangent line to a curve.

How do you find the derivative of a function?

To find the derivative of a function, you can use the rules of differentiation, such as the power rule, product rule, quotient rule, and chain rule. These rules involve taking the derivative of each term in the function and combining them using algebra.

Why is finding derivatives important?

Finding derivatives is important because it allows us to understand the behavior of a function and how it changes over time. It is also used in many real-world applications, such as physics, economics, and engineering.

Can derivatives be negative?

Yes, derivatives can be negative. This means that the function is decreasing at that point, and the slope of the tangent line is negative.

What is the difference between a derivative and an antiderivative?

A derivative represents the rate of change of a function, while an antiderivative represents the original function before differentiation. In other words, the antiderivative is the reverse process of finding the derivative.

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