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Evaluate the integral #int \ x^2(x^3-1)^4 \ dx #?

Answer 1

Charles Arocha

# int \ x^2(x^3-1)^4 \ dx = 1/15(x^3-1)^5 + C #

We want to find:

# I = int \ x^2(x^3-1)^4 \ dx #

We can perform a simple substitution; Let

# u = x^3-1 => (du)/dx = 3x^2 #

If we perform the substitution then we get:

# I = int \ 1/3u^4 \ du #

So we can now integrate to get:

# I = 1/3*1/5u^5 + C # # \ \ = 1/15 u^5 + C #

And restoring the substitution we get:

# I = 1/15(x^3-1)^5 + C #

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Answer 2

Elijah Abram

To evaluate the integral ( \int x^2(x^3-1)^4 , dx ):

Let ( u = x^3 - 1 ). Then, ( du = 3x^2 , dx ).

Substitute ( u ) and ( du ) into the integral:

[ \frac{1}{3} \int u^4 , du ]

Integrate ( u^4 ) with respect to ( u ):

[ \frac{1}{3} \left( \frac{u^5}{5} \right) + C ]

Substitute back ( u = x^3 - 1 ):

[ \frac{1}{15} (x^3 - 1)^5 + C ]

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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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