Find the Integral of x^3 * (x^2 - 8)^1/2 | Calculus 2

In summary, to find the integral of x^3 * ( x^2 - 8 )^1/2, you can use multiple methods such as integration by parts, trigonometric substitution, or hyperbolic substitution. However, the most efficient approach would be to use substitution with the suggested values, resulting in the answer of 1/15(x^2-8)^3/2 * (3x^2 + 16 ).
  • #1
waterchan
23
0
How would you find the integral of x^3 * ( x^2 - 8 )^1/2 ?

I've tried everything I can think of; substitution / trigonometric substitution, partial fractions, integration by parts, but I just can't get the answer. The answer is supposed to be:

1/15(x^2-8)^3/2 * (3x^2 + 16 )
 
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  • #2
Separate x^3 into x^2*x, then do integration by parts with x^2 for one and x(x^2-8)^.5 for the other.
 
  • #3
You could try a hyperbolic substitution.

[tex] \frac{x}{2\sqrt{2}}=\cosh t [/tex]

Daniel.
 
  • #4
Or just let u = x2 - 8, so x^3dx = (u+8)du/2.

--J
 
  • #5
Or you can do trigonometric substitution.

[tex] x = \sqrt{8} sec\theta; [/tex]

[tex] dx = \sqrt{8} sec\theta tan\theta d\theta ;[/tex]

and [tex] x^3 = 8\sqrt{8} sec^3\theta [/tex]

Edit message:

Blah i would definitely go for Justin's substitutions.
 
Last edited:
  • #6
Trig substitution will definitely work, you can simplify towards the step

[tex]64 \sqrt{(8)} \int sec^{4} \theta~tan^{2} \thetad \theta [/tex]

[tex]64 \sqrt{(8)} \int sec^{2} \theta~(1+tan^{2} \theta )tan^{2} \theta d \theta [/tex]

[tex]u=tan \theta [/tex]

[tex]du= sec^{2} \theta d \theta [/tex]
 

Related to Find the Integral of x^3 * (x^2 - 8)^1/2 | Calculus 2

1. What is the formula for finding the integral of x^3 * (x^2 - 8)^1/2?

The formula for finding the integral of x^3 * (x^2 - 8)^1/2 is ∫x^3 * (x^2 - 8)^1/2 dx = (1/5)(x^2 - 8)^3/2 + C.

2. What is the power rule in calculus?

The power rule in calculus states that the integral of x^n is equal to (1/(n+1))x^(n+1) + C.

3. How do you solve integrals using the substitution method?

To solve an integral using the substitution method, you must first identify a u-substitution, where u is a function of x. Then, use the formula ∫f(g(x))g'(x)dx = ∫f(u)du to rewrite the integral in terms of u. Finally, solve for the integral in terms of u and then replace u with the original function of x.

4. Can you use the chain rule to solve this integral?

No, the chain rule is used to find the derivative of a composite function, not the integral.

5. What is the purpose of finding the integral of a function?

The purpose of finding the integral of a function is to calculate the area under the curve of the function. This can be useful in many real-world applications, such as calculating the distance traveled by an object or finding the volume of an irregular shape.

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