How do you integrate #(x^3cos(x^2)dx)#?
I found:
I used Substitution and Integration by Parts:
By signing up, you agree to our Terms of Service and Privacy Policy
To integrate ( x^3 \cos(x^2) , dx ), you can use a substitution method. Let ( u = x^2 ), then ( du = 2x , dx ). Solving for ( dx ), we have ( dx = \frac{du}{2x} ). Substituting these into the integral, we get:
[ \int x^3 \cos(x^2) , dx = \int \frac{u}{2} \cos(u) , du ]
This can be simplified further by using integration by parts. Let ( dv = \cos(u) , du ), then ( v = \sin(u) ). Applying the integration by parts formula:
[ \int u , dv = uv - \int v , du ]
We get:
[ \frac{u \sin(u)}{2} - \int \frac{\sin(u)}{2} , du ]
Integrating ( \frac{\sin(u)}{2} ), we get:
[ \frac{u \sin(u)}{2} + \frac{\cos(u)}{2} + C ]
Finally, replacing ( u ) back with ( x^2 ), we have:
[ \frac{x^2 \sin(x^2)}{2} + \frac{\cos(x^2)}{2} + C ]
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.
- What is #f(x) = int e^(2x)-e^x+x dx# if #f(4 ) = 2 #?
- How do you integrate #int (3x-4)/((x-1)(x+3)(x-6)) # using partial fractions?
- How do you integrate #int x /sqrt( 81 - x^4 )dx# using trigonometric substitution?
- How do you integrate #e^2xcosx dx#?
- How do you integrate #int x /sqrt(1 + x^2) dx# using trigonometric substitution?
- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7