Find definite integral. ∫02x2(3x3+1)13dx

Joan Thompson

Joan Thompson

Answered

2021-12-31

Find definite integral.
02x2(3x3+1)13dx

Answer & Explanation

Jenny Sheppard

Jenny Sheppard

Expert

2022-01-01Added 35 answers

Step 1
The given integral can be solved by the method of substitution. The substitution that will be used is
3x3+1=u. This gives 9x2dx=du or x2dx=du9.
This substitution absorbs the x2dx term into du9. Calculate the corresponding limits of the integration in terms of the new variable u.
Step 2
Limits of integration for x are from 0 to 2. New variable for integration is u=3x3+1. So the lower limit of integration in terms of new variable will be 1. Calculate the upper limit by substituting x=2.
u=323+1
=3*8+1
=25
So, the integration with this substitution becomes 125u13du9. Calculate this integral using the integral
xndx=xn+1n+1.
125u13du9=19(u13+113+1)125
=1934(u43)125
=112(25431)
Hence, the given definite integral is equal to 112(25431)
Shawn Kim

Shawn Kim

Expert

2022-01-02Added 25 answers

02x2(3x3+1)1/3dx=x23x3+13dx
=19u3du
u3du
=3u434
19u3du
=u4312
=(3x3+1)4312
x23x3+13
=(3x3+1)4312+C

Vasquez

Vasquez

Expert

2022-01-07Added 457 answers

20x2(3x3+1)1/3dxx2×(3x3+1)1/3dxt139dt19×t13dt19×3tt3419×3(3x3+1)3x3+134(3x3+1)3x3+1312(3x3+1)3x3+1312|02(3×23+1)3×23+1312(3×03+1)3×03+1312Answer:25253112

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