How would I find the Integral of
3dx/(10x^2-20x+20)
I've been stuck on this, and would like an answer ASAP.
try factoring the bottom and using the triangle rule
There's my problem. I just can't factor the bottom.
Actually, it's arctan.
3/10*integral(1/((x-1)^2+1))dx
=3/10arctan(x-1) + C
Oh. Okay thanks.
Thank you so much. If it's not to much...
How would I integrate
(4dt)/(sqrt(15-2t-t^2))
That thing has me stumped.
that one's arcsin.
factor it into:
4*integral(1/sqrt(16-(t^2+2t+1)))dt
so,
4*integral(1/sqrt(4^2+(t+1)^2))dt
divide everything in the sqrt by 4^2 and bring it out as 1/4,
4/4*integral(1/sqrt(1+((t+1)/4)^s))dt)
now it's in arcsin form (remember u and du)
so your answer is:
4*arcsin((t+1)/4) + C
To find the integral of
∫(3dx / (10x^2 - 20x + 20)),
we can start by factoring the denominator, if possible. However, in this case, the denominator cannot be easily factored.
Since factoring the denominator does not work, an alternative approach for integrating this type of rational function is to use partial fraction decomposition.
Here are the steps to find the partial fraction decomposition:
1. Start with the integrand and split it into partial fractions:
(3dx / (10x^2 - 20x + 20)) = (A / (x^2 - 2x + 2)).
2. Multiply both sides of the equation by the denominator of the integrand (x^2 - 2x + 2) to eliminate the fraction:
3dx = A(x^2 - 2x + 2)dx.
3. Expand the equation:
3dx = (Ax^2 - 2Ax + 2A)dx.
4. Equate the coefficients of the terms on both sides:
The coefficient of x^2 on the left-hand side is 0, and on the right-hand side is A. Therefore, A = 0.
The coefficient of x on the left-hand side is 3, and on the right-hand side is -2A. Therefore, -2A = 3, or A = -3/2.
The constant term on the left-hand side is 0, and on the right-hand side is 2A. Therefore, 2A = 0, or A = 0.
5. Rewrite the original integral using the partial fractions:
∫ (3dx / (10x^2 - 20x + 20)) = ∫ (0dx / (x^2 - 2x + 2)).
Now that we have rewritten the integral using partial fractions, we can proceed to integrate it.
The integral of 0dx is simply 0.
Therefore, the integral of (3dx / (10x^2 - 20x + 20)) is 0 + C, where C is the constant of integration.
Thus, the final answer is ∫ (3dx / (10x^2 - 20x + 20)) = C.