How do you integrate #int x^3*(1-x^4)^6 dx #?

Answer 1

#intx^3(1-x^4)^6dx=-1/28(1-x^4)^7+c#

#intx^3(1-x^4)^6dx---(1)#

method 1: inspection

we note that the function outside the bracket is a multiple of the bracket differentiated, so we can approach this integral by inspection.

We will 'guess' the bracket to the power +1, differentiate and compare it with the required integral.

#d/(dx)(1-x^4)^7=7(1-x^4)^6xx(-4x^3)#
#=-28x^3(1-x^4)^6--(2)#
comparing #(1) " & "(2)# we can conclude
#intx^3(1-x^4)^6dx=-1/28(1-x^4)^7+c#

method 2 : substitution

#intx^3(1-x^4)^6dx---(1)#
#u=1-x^4=>du=-4x^3dx#
#(1)rarrintcancel(x^3)u^6xx-1/(4cancel(x^3))du#
#=-1/4intu^6du=-1/4xxu^7/7+c#
#=-1/28u^7+c=-1/28(1-x^4)^7+c#
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Answer 2

To integrate ( \int x^3(1-x^4)^6 , dx ), you can use the substitution method. Let ( u = 1 - x^4 ), then ( du = -4x^3 , dx ). Solving for ( dx ), we get ( dx = -\frac{1}{4x^3} , du ).

Now, substitute ( u = 1 - x^4 ) and ( dx = -\frac{1}{4x^3} , du ) into the integral:

[ \int x^3(1-x^4)^6 , dx = \int -\frac{1}{4x^3} u^6 , du ]

[ = -\frac{1}{4} \int \frac{u^6}{x^3} , du ]

Now, we need to express ( x^3 ) in terms of ( u ). From the substitution ( u = 1 - x^4 ), we can rearrange it to solve for ( x^4 ), then take the fourth root:

[ u = 1 - x^4 ]

[ x^4 = 1 - u ]

[ x = (1 - u)^{\frac{1}{4}} ]

[ x^3 = (1 - u)^{\frac{3}{4}} ]

Substitute this expression into the integral:

[ = -\frac{1}{4} \int \frac{u^6}{(1 - u)^{\frac{3}{4}}} , du ]

Now, we can use a standard integration technique or software to solve this integral. Once integrated, we can express the result back in terms of ( x ).

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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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