How do you evaluate the integral #intx^nlnxdx#?
For For
Employ integration in segments. Let
Then:
By signing up, you agree to our Terms of Service and Privacy Policy
To evaluate the integral (\int x^n \ln x , dx), you can use integration by parts. The integration by parts formula states:
[ \int u , dv = uv - \int v , du ]
For this integral, let ( u = \ln x ) and ( dv = x^n , dx ). Then, differentiate ( u ) to find ( du ) and integrate ( dv ) to find ( v ). The steps are as follows:
- Let ( u = \ln x ) and ( dv = x^n , dx ).
- Differentiate ( u ) to find ( du ): ( du = \frac{1}{x} , dx ).
- Integrate ( dv ) to find ( v ): ( v = \frac{x^{n+1}}{n+1} ).
- Apply the integration by parts formula:
[ \int x^n \ln x , dx = uv - \int v , du ] [ = \ln x \cdot \frac{x^{n+1}}{n+1} - \int \frac{x^{n+1}}{n+1} \cdot \frac{1}{x} , dx ] [ = \frac{x^{n+1} \ln x}{n+1} - \int \frac{x^n}{n+1} , dx ] [ = \frac{x^{n+1} \ln x}{n+1} - \frac{1}{n+1} \int x^n , dx ]
Now integrate ( \int x^n , dx ) separately:
[ \int x^n , dx = \frac{x^{n+1}}{n+1} + C ]
Substitute this back into the previous equation:
[ \int x^n \ln x , dx = \frac{x^{n+1} \ln x}{n+1} - \frac{1}{n+1} \left( \frac{x^{n+1}}{n+1} + C \right) ]
Simplify to get the final result:
[ \int x^n \ln x , dx = \frac{x^{n+1} \ln x}{n+1} - \frac{x^{n+1}}{(n+1)^2} - \frac{C}{n+1} ]
where ( C ) is the constant of integration.
By signing up, you agree to our Terms of Service and Privacy Policy
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.
- How do you evaluate the integral #int x^2lnx dx# from 0 to 1 if it converges?
- What is the antiderivative of #(sinx)*(cosx)#?
- What is the net area between #f(x) = x^2+1/x # and the x-axis over #x in [2, 4 ]#?
- How do you find the antiderivative of #e^(2x)/sqrt(1-e^x)#?
- How do you evaluate the definite integral #int 4-x# from #[0,2]#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7