How do you integrate #int x^3lnabsx# from 1 to 2 by integration by parts method?
By signing up, you agree to our Terms of Service and Privacy Policy
To integrate ( \int x^3 \ln x  , dx ) from 1 to 2 using integration by parts:
 Let ( u = \ln x  ) and ( dv = x^3 , dx ).
 Find ( du ) and ( v ).
 Apply integration by parts formula ( \int u , dv = uv  \int v , du ).
 Calculate the definite integral from 1 to 2.
Here are the steps:

Let ( u = \ln x  ) and ( dv = x^3 , dx ). Then, ( du = \frac{1}{x} , dx ) and ( v = \frac{x^4}{4} ).

Apply integration by parts formula: [ \int x^3 \ln x  , dx = \frac{x^4 \ln x }{4}  \int \frac{x^4}{4} \cdot \frac{1}{x} , dx ]

Simplify the integral: [ = \frac{x^4 \ln x }{4}  \frac{1}{4} \int x^3 , dx ]

Integrate the remaining integral: [ = \frac{x^4 \ln x }{4}  \frac{1}{4} \cdot \frac{x^4}{4} \bigg_1^2 ]

Evaluate the definite integral: [ = \left( \frac{2^4 \ln 2 }{4}  \frac{1}{4} \cdot \frac{2^4}{4} \right)  \left( \frac{1^4 \ln 1 }{4}  \frac{1}{4} \cdot \frac{1^4}{4} \right) ]

Simplify and compute: [ = \left( \frac{16 \ln 2 }{4}  \frac{16}{4} \right)  \left( \frac{0}{4}  \frac{1}{16} \right) ] [ = \left( 4 \ln 2   4 \right)  \left(  \frac{1}{16} \right) ] [ = 4 \ln 2  4 + \frac{1}{16} ]
So, the integral of ( \int_{1}^{2} x^3 \ln x  , dx ) is ( 4 \ln 2  4 + \frac{1}{16} ).
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a onesided 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 onesided 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 onesided 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 onesided 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.
 How to know when to use integration by substitution vs. integration by parts?
 How do you integrate #f(x)=(3x^2x)/((x^252)(x+4)(x7))# using partial fractions?
 What is the integral of 5(6t+e^(t))(2t+4) with respect to t?
 How do you integrate #xln(1+x) dx#?
 How do you evaluate the integral #int arctansqrtx#?
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7