How do you evaluate the definite integral #int sinsqrtx dx# from #[0, pi^2]#?
Let's integrate without bounds first.
Thus:
We can now use the bounds:
By signing up, you agree to our Terms of Service and Privacy Policy
To evaluate the definite integral ( \int_{0}^{\pi^2} \sin(\sqrt{x}) , dx ), you can use a substitution method. Let ( u = \sqrt{x} ), then ( du = \frac{1}{2\sqrt{x}} dx ). Rearranging, we find ( dx = 2u , du ).
Now, when ( x = 0 ), ( u = \sqrt{0} = 0 ), and when ( x = \pi^2 ), ( u = \sqrt{\pi^2} = \pi ).
So, the integral becomes:
[ \int_{0}^{\pi} \sin(u) \cdot 2u , du ]
Now, integrate ( \sin(u) \cdot 2u ):
[ \int_{0}^{\pi} 2u \sin(u) , du ]
To evaluate this integral, you can use integration by parts:
[ u \cdot dv = 2u \sin(u) , du ] [ du = du ] [ dv = \sin(u) , du ] [ v = -\cos(u) ]
Applying integration by parts:
[ \int 2u \sin(u) , du = -2u \cos(u) - \int -2\cos(u) , du ] [ = -2u \cos(u) + 2 \sin(u) + C ]
Now, apply the limits of integration:
[ -2\pi \cos(\pi) + 2 \sin(\pi) - (-2\cdot 0 \cos(0) + 2 \sin(0)) ] [ = 2\pi ]
So, ( \int_{0}^{\pi^2} \sin(\sqrt{x}) , dx = 2\pi ).
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 definite integral #int (x-x^3)dx# from [0,1]?
- How do you find the integral of #(x^4+x-4) / (x^2+2)#?
- How do you find the partial sum of #Sigma (2n-1)# from n=1 to 400?
- How do you differentiate #G(x) = intsqrtt sint dt# from #sqrt(x)# to #x^3#?
- How do you evaluate the integral of #int ln(1+x^3)dx#?

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