How do you find the integral of #int sin^5(x)cos^8(x) dx#?

Answer 1

#-cos^9x/9 + (2cos^11x)/11 - cos^13x/13 + C#

#I=int sin^5x cos^8xdx=intsinx sin^4x cos^8x dx#
#I=int sinx (sin^2x)^2 cos^8xdx=int (1-cos^2x)^2 cos^8x sinx dx#
#cosx=t => -sinxdx=dt => sinxdx=-dt#
#I=int (1-t^2)^2 t^8 (-dt) = -int (1-2t^2+t^4)t^8 dt#
#I=-int (t^8-2t^10+t^12) dt = -t^9/9+(2t^11)/11-t^13/13+C#
#I=-cos^9x/9 + (2cos^11x)/11 - cos^13x/13 + C#
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Answer 2

To find the integral of (\int \sin^5(x)\cos^8(x) , dx), you can use trigonometric identities and integration by parts. Here's how:

  1. Start by using the identity (\sin^2(x) = 1 - \cos^2(x)) to rewrite (\sin^5(x)) as ((1 - \cos^2(x))^2\sin(x)).
  2. Expand ((1 - \cos^2(x))^2) using the binomial theorem.
  3. Now you have an integral involving powers of (\sin(x)) and (\cos(x)), which you can integrate using integration by parts.
  4. Let (u = \cos(x)) and (dv = \sin^4(x)\cos^4(x) , dx), then differentiate (u) to find (du) and integrate (dv) to find (v).
  5. Apply integration by parts formula: (\int u , dv = uv - \int v , du).
  6. Substitute the values of (u), (du), (dv), and (v) into the integration by parts formula and evaluate the integral.
  7. Repeat the process if necessary until you obtain an expression that can be easily integrated.

Following these steps will lead you to the solution of the integral (\int \sin^5(x)\cos^8(x) , dx).

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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.

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.

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.

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.

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