Evaluate the definite integral using integration by parts. ∫02z(z−2)4dz

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Evaluate the definite integral using integration by parts.  ∫02z(z−2)4dz

Salvador Fry · Accepted Answer

Step 1  We need to Evaluate the definite integral using integration by parts of the following integral.  ∫02z(z−2)4dz  Note:  Integration by parts:  ∫abvdudzdz=[uv−∫udvdzdz]ab  Step 2  So,  For the given integral,  ∫02z(z−2)4dz=[z(z−2)55−∫(z−2)55×ddz(z)dz]02  =[z(z−2)55−∫(z−2)55dz]02  =[z(z−2)55−(z−2)65×6]02  =0−(0−(0−2)65×6)  =3215  Answer:  In the exact form: ∫02z(z−2)4dz=[z(z−2)55−∫(z−2)55×ddz(z)dz]02  And the value of the integral is ∫02z(z−2)4dz=3215.

Marian Tucker · Accepted Answer

Step 1: If f(x) is a continuous function from a to b, and if F(x) is its integral, then:  ∫abf(x)dx=F(x)∣ab=F(b)−F(a)  Step 2: In this case, f(z)=z(z−2)4. Find its integral.  (z−2)66+2(z−2)55∣02  Step 3: Since F(z)∣ab=F(b)−F(a), expand the above into F(2)−F(0):  ((2−2)66+2(2−2)55)−((0−2)66+2(0−2)55)  Step 4: Simplify.  3215

Evaluate the definite integral using integration by parts. \int_{0}^{2}z(z-2)^{4}dz

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