Calculus Examples

Evaluate the Integral integral of 5x^2(3x-4)(x-1) with respect to x
5x2(3x-4)(x-1)dx
Step 1
Since 5 is constant with respect to x, move 5 out of the integral.
5(x2(3x-4))(x-1)dx
Step 2
Let u1=x-1. Then du1=dx. Rewrite using u1 and du1.
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Step 2.1
Let u1=x-1. Find du1dx.
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Step 2.1.1
Differentiate x-1.
ddx[x-1]
Step 2.1.2
By the Sum Rule, the derivative of x-1 with respect to x is ddx[x]+ddx[-1].
ddx[x]+ddx[-1]
Step 2.1.3
Differentiate using the Power Rule which states that ddx[xn] is nxn-1 where n=1.
1+ddx[-1]
Step 2.1.4
Since -1 is constant with respect to x, the derivative of -1 with respect to x is 0.
1+0
Step 2.1.5
Add 1 and 0.
1
1
Step 2.2
Rewrite the problem using u1 and du1.
5(u1+1)2(3(u1+1)-4)u1du1
5(u1+1)2(3(u1+1)-4)u1du1
Step 3
Let u2=u1+1. Then du2=du1. Rewrite using u2 and du2.
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Step 3.1
Let u2=u1+1. Find du2du1.
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Step 3.1.1
Differentiate u1+1.
ddu1[u1+1]
Step 3.1.2
By the Sum Rule, the derivative of u1+1 with respect to u1 is ddu1[u1]+ddu1[1].
ddu1[u1]+ddu1[1]
Step 3.1.3
Differentiate using the Power Rule which states that ddu1[u1n] is nu1n-1 where n=1.
1+ddu1[1]
Step 3.1.4
Since 1 is constant with respect to u1, the derivative of 1 with respect to u1 is 0.
1+0
Step 3.1.5
Add 1 and 0.
1
1
Step 3.2
Rewrite the problem using u2 and du2.
5u22(3u2-4)(u2-1)du2
5u22(3u2-4)(u2-1)du2
Step 4
Let u3=u2-1. Then du3=du2. Rewrite using u3 and du3.
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Step 4.1
Let u3=u2-1. Find du3du2.
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Step 4.1.1
Differentiate u2-1.
ddu2[u2-1]
Step 4.1.2
By the Sum Rule, the derivative of u2-1 with respect to u2 is ddu2[u2]+ddu2[-1].
ddu2[u2]+ddu2[-1]
Step 4.1.3
Differentiate using the Power Rule which states that ddu2[u2n] is nu2n-1 where n=1.
1+ddu2[-1]
Step 4.1.4
Since -1 is constant with respect to u2, the derivative of -1 with respect to u2 is 0.
1+0
Step 4.1.5
Add 1 and 0.
1
1
Step 4.2
Rewrite the problem using u3 and du3.
5(u3+1)2(3(u3+1)-4)u3du3
5(u3+1)2(3(u3+1)-4)u3du3
Step 5
Simplify.
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Step 5.1
Rewrite (u3+1)2 as (u3+1)(u3+1).
5(u3+1)(u3+1)(3(u3+1)-4)u3du3
Step 5.2
Apply the distributive property.
5(u3(u3+1)+1(u3+1))(3(u3+1)-4)u3du3
Step 5.3
Apply the distributive property.
5(u3u3+u31+1(u3+1))(3(u3+1)-4)u3du3
Step 5.4
Apply the distributive property.
5(u3u3+u31+1u3+11)(3(u3+1)-4)u3du3
Step 5.5
Apply the distributive property.
5(u3u3+u31+1u3+11)(3u3+31-4)u3du3
Step 5.6
Apply the distributive property.
5((u3u3+u31)(3u3+31-4)+(1u3+11)(3u3+31-4))u3du3
Step 5.7
Apply the distributive property.
5(u3u3(3u3+31-4)+u31(3u3+31-4)+(1u3+11)(3u3+31-4))u3du3
Step 5.8
Apply the distributive property.
5(u3u3(3u3+31)+u3u3-4+u31(3u3+31-4)+(1u3+11)(3u3+31-4))u3du3
Step 5.9
Apply the distributive property.
5(u3u3(3u3)+u3u3(31)+u3u3-4+u31(3u3+31-4)+(1u3+11)(3u3+31-4))u3du3
Step 5.10
Apply the distributive property.
5(u3u3(3u3)+u3u3(31)+u3u3-4+u31(3u3+31)+u31-4+(1u3+11)(3u3+31-4))u3du3
Step 5.11
Apply the distributive property.
5(u3u3(3u3)+u3u3(31)+u3u3-4+u31(3u3)+u31(31)+u31-4+(1u3+11)(3u3+31-4))u3du3
Step 5.12
Apply the distributive property.
5(u3u3(3u3)+u3u3(31)+u3u3-4+u31(3u3)+u31(31)+u31-4+1u3(3u3+31-4)+11(3u3+31-4))u3du3
Step 5.13
Apply the distributive property.
5(u3u3(3u3)+u3u3(31)+u3u3-4+u31(3u3)+u31(31)+u31-4+1u3(3u3+31)+1u3-4+11(3u3+31-4))u3du3
Step 5.14
Apply the distributive property.
5(u3u3(3u3)+u3u3(31)+u3u3-4+u31(3u3)+u31(31)+u31-4+1u3(3u3)+1u3(31)+1u3-4+11(3u3+31-4))u3du3
Step 5.15
Apply the distributive property.
5(u3u3(3u3)+u3u3(31)+u3u3-4+u31(3u3)+u31(31)+u31-4+1u3(3u3)+1u3(31)+1u3-4+11(3u3+31)+11-4)u3du3
Step 5.16
Apply the distributive property.
5(u3u3(3u3)+u3u3(31)+u3u3-4+u31(3u3)+u31(31)+u31-4+1u3(3u3)+1u3(31)+1u3-4+11(3u3)+11(31)+11-4)u3du3
Step 5.17
Apply the distributive property.
5(u3u3(3u3)+u3u3(31)+u3u3-4+u31(3u3)+u31(31)+u31-4)u3+(1u3(3u3)+1u3(31)+1u3-4+11(3u3)+11(31)+11-4)u3du3
Step 5.18
Apply the distributive property.
5(u3u3(3u3)+u3u3(31)+u3u3-4)u3+(u31(3u3)+u31(31)+u31-4)u3+(1u3(3u3)+1u3(31)+1u3-4+11(3u3)+11(31)+11-4)u3du3
Step 5.19
Apply the distributive property.
5(u3u3(3u3)+u3u3(31))u3+u3u3-4u3+(u31(3u3)+u31(31)+u31-4)u3+(1u3(3u3)+1u3(31)+1u3-4+11(3u3)+11(31)+11-4)u3du3
Step 5.20
Apply the distributive property.
5u3u3(3u3)u3+u3u3(31)u3+u3u3-4u3+(u31(3u3)+u31(31)+u31-4)u3+(1u3(3u3)+1u3(31)+1u3-4+11(3u3)+11(31)+11-4)u3du3
Step 5.21
Apply the distributive property.
5u3u3(3u3)u3+u3u3(31)u3+u3u3-4u3+(u31(3u3)+u31(31))u3+u31-4u3+(1u3(3u3)+1u3(31)+1u3-4+11(3u3)+11(31)+11-4)u3du3
Step 5.22
Apply the distributive property.
5u3u3(3u3)u3+u3u3(31)u3+u3u3-4u3+u31(3u3)u3+u31(31)u3+u31-4u3+(1u3(3u3)+1u3(31)+1u3-4+11(3u3)+11(31)+11-4)u3du3
Step 5.23
Apply the distributive property.
5u3u3(3u3)u3+u3u3(31)u3+u3u3-4u3+u31(3u3)u3+u31(31)u3+u31-4u3+(1u3(3u3)+1u3(31)+1u3-4)u3+(11(3u3)+11(31)+11-4)u3du3
Step 5.24
Apply the distributive property.
5u3u3(3u3)u3+u3u3(31)u3+u3u3-4u3+u31(3u3)u3+u31(31)u3+u31-4u3+(1u3(3u3)+1u3(31))u3+1u3-4u3+(11(3u3)+11(31)+11-4)u3du3
Step 5.25
Apply the distributive property.
5u3u3(3u3)u3+u3u3(31)u3+u3u3-4u3+u31(3u3)u3+u31(31)u3+u31-4u3+1u3(3u3)u3+1u3(31)u3+1u3-4u3+(11(3u3)+11(31)+11-4)u3du3
Step 5.26
Apply the distributive property.
5u3u3(3u3)u3+u3u3(31)u3+u3u3-4u3+u31(3u3)u3+u31(31)u3+u31-4u3+1u3(3u3)u3+1u3(31)u3+1u3-4u3+(11(3u3)+11(31))u3+11-4u3du3
Step 5.27
Apply the distributive property.
5u3u3(3u3)u3+u3u3(31)u3+u3u3-4u3+u31(3u3)u3+u31(31)u3+u31-4u3+1u3(3u3)u3+1u3(31)u3+1u3-4u3+11(3u3)u3+11(31)u3+11-4u3du3
Step 5.28
Move u3.
5u33u3u3u3+u3u3(31)u3+u3u3-4u3+u31(3u3)u3+u31(31)u3+u31-4u3+1u3(3u3)u3+1u3(31)u3+1u3-4u3+11(3u3)u3+11(31)u3+11-4u3du3
Step 5.29
Reorder u3 and 3.
53u3u3u3u3+u3u3(31)u3+u3u3-4u3+u31(3u3)u3+u31(31)u3+u31-4u3+1u3(3u3)u3+1u3(31)u3+1u3-4u3+11(3u3)u3+11(31)u3+11-4u3du3
Step 5.30
Move u3.
53u3u3u3u3+u331u3u3+u3u3-4u3+u31(3u3)u3+u31(31)u3+u31-4u3+1u3(3u3)u3+1u3(31)u3+1u3-4u3+11(3u3)u3+11(31)u3+11-4u3du3
Step 5.31
Move u3.
53u3u3u3u3+31u3u3u3+u3u3-4u3+u31(3u3)u3+u31(31)u3+u31-4u3+1u3(3u3)u3+1u3(31)u3+1u3-4u3+11(3u3)u3+11(31)u3+11-4u3du3
Step 5.32
Move u3.
53u3u3u3u3+31u3u3u3+u3-4u3u3+u31(3u3)u3+u31(31)u3+u31-4u3+1u3(3u3)u3+1u3(31)u3+1u3-4u3+11(3u3)u3+11(31)u3+11-4u3du3
Step 5.33
Reorder u3 and -4.
53u3u3u3u3+31u3u3u3-4u3u3u3+u31(3u3)u3+u31(31)u3+u31-4u3+1u3(3u3)u3+1u3(31)u3+1u3-4u3+11(3u3)u3+11(31)u3+11-4u3du3
Step 5.34
Reorder u3 and 1.
53u3u3u3u3+31u3u3u3-4u3u3u3+1u3(3u3)u3+u31(31)u3+u31-4u3+1u3(3u3)u3+1u3(31)u3+1u3-4u3+11(3u3)u3+11(31)u3+11-4u3du3
Step 5.35
Move u3.
53u3u3u3u3+31u3u3u3-4u3u3u3+13u3u3u3+u31(31)u3+u31-4u3+1u3(3u3)u3+1u3(31)u3+1u3-4u3+11(3u3)u3+11(31)u3+11-4u3du3
Step 5.36
Reorder u3 and 1.
53u3u3u3u3+31u3u3u3-4u3u3u3+13u3u3u3+1u3(31)u3+u31-4u3+1u3(3u3)u3+1u3(31)u3+1u3-4u3+11(3u3)u3+11(31)u3+11-4u3du3
Step 5.37
Move u3.
53u3u3u3u3+31u3u3u3-4u3u3u3+13u3u3u3+131u3u3+u31-4u3+1u3(3u3)u3+1u3(31)u3+1u3-4u3+11(3u3)u3+11(31)u3+11-4u3du3
Step 5.38
Reorder u3 and 1.
53u3u3u3u3+31u3u3u3-4u3u3u3+13u3u3u3+131u3u3+1u3-4u3+1u3(3u3)u3+1u3(31)u3+1u3-4u3+11(3u3)u3+11(31)u3+11-4u3du3
Step 5.39
Move u3.
53u3u3u3u3+31u3u3u3-4u3u3u3+13u3u3u3+131u3u3+1-4u3u3+1u3(3u3)u3+1u3(31)u3+1u3-4u3+11(3u3)u3+11(31)u3+11-4u3du3
Step 5.40
Move u3.
53u3u3u3u3+31u3u3u3-4u3u3u3+13u3u3u3+131u3u3+1-4u3u3+13u3u3u3+1u3(31)u3+1u3-4u3+11(3u3)u3+11(31)u3+11-4u3du3
Step 5.41
Move u3.
53u3u3u3u3+31u3u3u3-4u3u3u3+13u3u3u3+131u3u3+1-4u3u3+13u3u3u3+131u3u3+1u3-4u3+11(3u3)u3+11(31)u3+11-4u3du3
Step 5.42
Move u3.
53u3u3u3u3+31u3u3u3-4u3u3u3+13u3u3u3+131u3u3+1-4u3u3+13u3u3u3+131u3u3+1-4u3u3+11(3u3)u3+11(31)u3+11-4u3du3
Step 5.43
Raise u3 to the power of 1.
53(u31u3)u3u3+31u3u3u3-4u3u3u3+13u3u3u3+131u3u3+1-4u3u3+13u3u3u3+131u3u3+1-4u3u3+113u3u3+1131u3+11-4u3du3
Step 5.44
Raise u3 to the power of 1.
53(u31u31)u3u3+31u3u3u3-4u3u3u3+13u3u3u3+131u3u3+1-4u3u3+13u3u3u3+131u3u3+1-4u3u3+113u3u3+1131u3+11-4u3du3
Step 5.45
Use the power rule aman=am+n to combine exponents.
53u31+1u3u3+31u3u3u3-4u3u3u3+13u3u3u3+131u3u3+1-4u3u3+13u3u3u3+131u3u3+1-4u3u3+113u3u3+1131u3+11-4u3du3
Step 5.46
Add 1 and 1.
53u32u3u3+31u3u3u3-4u3u3u3+13u3u3u3+131u3u3+1-4u3u3+13u3u3u3+131u3u3+1-4u3u3+113u3u3+1131u3+11-4u3du3
Step 5.47
Raise u3 to the power of 1.
53(u32u31)u3+31u3u3u3-4u3u3u3+13u3u3u3+131u3u3+1-4u3u3+13u3u3u3+131u3u3+1-4u3u3+113u3u3+1131u3+11-4u3du3
Step 5.48
Use the power rule to combine exponents.
Step 5.49
Add and .
Step 5.50
Raise to the power of .
Step 5.51
Use the power rule to combine exponents.
Step 5.52
Add and .
Step 5.53
Multiply by .
Step 5.54
Raise to the power of .
Step 5.55
Raise to the power of .
Step 5.56
Use the power rule to combine exponents.
Step 5.57
Add and .
Step 5.58
Raise to the power of .
Step 5.59
Use the power rule to combine exponents.
Step 5.60
Add and .
Step 5.61
Raise to the power of .
Step 5.62
Raise to the power of .
Step 5.63
Use the power rule to combine exponents.
Step 5.64
Add and .
Step 5.65
Raise to the power of .
Step 5.66
Use the power rule to combine exponents.
Step 5.67
Add and .
Step 5.68
Subtract from .
Step 5.69
Multiply by .
Step 5.70
Raise to the power of .
Step 5.71
Raise to the power of .
Step 5.72
Use the power rule to combine exponents.
Step 5.73
Add and .
Step 5.74
Raise to the power of .
Step 5.75
Use the power rule to combine exponents.
Step 5.76
Add and .
Step 5.77
Multiply by .
Step 5.78
Multiply by .
Step 5.79
Raise to the power of .
Step 5.80
Raise to the power of .
Step 5.81
Use the power rule to combine exponents.
Step 5.82
Add and .
Step 5.83
Multiply by .
Step 5.84
Raise to the power of .
Step 5.85
Raise to the power of .
Step 5.86
Use the power rule to combine exponents.
Step 5.87
Add and .
Step 5.88
Subtract from .
Step 5.89
Add and .
Step 5.90
Multiply by .
Step 5.91
Raise to the power of .
Step 5.92
Raise to the power of .
Step 5.93
Use the power rule to combine exponents.
Step 5.94
Add and .
Step 5.95
Raise to the power of .
Step 5.96
Use the power rule to combine exponents.
Step 5.97
Add and .
Step 5.98
Multiply by .
Step 5.99
Multiply by .
Step 5.100
Raise to the power of .
Step 5.101
Raise to the power of .
Step 5.102
Use the power rule to combine exponents.
Step 5.103
Add and .
Step 5.104
Multiply by .
Step 5.105
Raise to the power of .
Step 5.106
Raise to the power of .
Step 5.107
Use the power rule to combine exponents.
Step 5.108
Add and .
Step 5.109
Subtract from .
Step 5.110
Multiply by .
Step 5.111
Multiply by .
Step 5.112
Raise to the power of .
Step 5.113
Raise to the power of .
Step 5.114
Use the power rule to combine exponents.
Step 5.115
Add and .
Step 5.116
Multiply by .
Step 5.117
Multiply by .
Step 5.118
Multiply by .
Step 5.119
Multiply by .
Step 5.120
Multiply by .
Step 5.121
Subtract from .
Step 5.122
Add and .
Step 5.123
Move .
Step 5.124
Move .
Step 5.125
Add and .
Step 5.126
Subtract from .
Step 6
Split the single integral into multiple integrals.
Step 7
Since is constant with respect to , move out of the integral.
Step 8
By the Power Rule, the integral of with respect to is .
Step 9
Since is constant with respect to , move out of the integral.
Step 10
By the Power Rule, the integral of with respect to is .
Step 11
By the Power Rule, the integral of with respect to is .
Step 12
Simplify.
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Step 12.1
Combine and .
Step 12.2
Combine and .
Step 12.3
Combine and .
Step 13
Since is constant with respect to , move out of the integral.
Step 14
By the Power Rule, the integral of with respect to is .
Step 15
Simplify.
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Step 15.1
Combine and .
Step 15.2
Simplify.
Step 16
Substitute back in for each integration substitution variable.
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Step 16.1
Replace all occurrences of with .
Step 16.2
Replace all occurrences of with .
Step 16.3
Replace all occurrences of with .
Step 17
Reorder terms.