Hello Everyone,
I need help with Calc II.
1. Integral from 0 to 1 of
(sin(3*pi*t))dt
For this one, I got -1/3pi cos (9 pi^2) + 1/3pi
2. indefinite integral of
sinxcos(cosx)dx
I got sin(cosx) + C
3. Indefinite integral of
x over (root (1-x^4))dx
I don't know how to solve this. Can I take U = root 1-x^4 or without the root?
Thank you for your help!
First is way wrong.
INT sin 3PIt= - 1/3PI * cos 3PI t
Second is right.
Third. I don't see it either. Will think on it, if I see a solution, I will post it later.
3. Indefinite integral of
x over (root (1-x^4))dx
Substitute x = sqrt[sin(t)]
You find that the indefinite integral in terms of t is:
1/2 t + c
So, in terms of x it is:
1/2 arcsin(x^2) + c
For the first question, you made an error in your integration. The correct answer for the integral from 0 to 1 of sin(3πt)dt is actually (-1/3π) * cos(3π) + (1/3π).
To solve this integral, you can use the power rule for integration:
∫sin(3πt)dt = (-1/3π) * cos(3πt) + C
Evaluate the antiderivative at the upper limit (1) and lower limit (0) to find the definite integral.
For the second question, you are correct. The indefinite integral of sin(x)cos(cos(x))dx is simply sin(cos(x)) + C, where C is the constant of integration.
Now, let's move on to the third question:
For the indefinite integral of x/(√(1-x^4))dx, you can use substitution to simplify the integral.
Let u = √(1 - x^4), then find du/dx by differentiating both sides:
du/dx = -2x^3 / (√(1 - x^4))
Rearrange the equation to solve for dx:
dx = - (√(1 - x^4)) * (1 / (2x^3)) * du
Substitute this value of dx into the integral:
∫(x / (√(1 - x^4))) dx = ∫((-√(1 - x^4)) * (1 / (2x^3))) du
Now, substitute u = √(1 - x^4) and dx = - (√(1 - x^4)) * (1 / (2x^3)) * du into the integral:
∫((-√(1 - x^4)) * (1 / (2x^3))) du = ∫(-1/(2x^3)) du
The new integral is simpler to solve. Integrate (-1 / (2x^3)) with respect to u:
= (-1/2) ∫(1 / (x^3)) du
Now, integrate (1 / (x^3)) with respect to u:
= (-1/2) * (-1 / (2x^2)) + C
= 1/(4x^2) + C
So, the indefinite integral of x/(√(1-x^4))dx is 1/(4x^2) + C.
Remember to always check your answer by differentiating it to see if it matches the original function!