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1. Integrals using long division

Homework Statement ∫(2x+1)/(x²+2x+1)(x²+x+1) Homework Equations none The Attempt at a Solution I've foiled this out to look like: ∫(2x+1)/(x^4+3x³+4x²+3x+1) I'm trying long division here but it's getting really ugly really fast. Should I foil this out in the first place or...

Homework Statement for the following integrals, am I allowed to break them up like so: 1. ∫(1)/(sqrt(16-9x²)³) dx = ∫(1)/(√16)³ · ∫(1)/(√-9x²)³ dx 2. ∫(x²)/(sqrt(x²-9)) dx = ∫(x²)/(√x²) · ∫(x²)/(√-9) dx 3. ∫(1)/(x²(sqrt(a²+x²))) dx = ∫(1)/(x²) · ∫(1)/(√a²) · ∫(1)/(√x²) dx...
3. The power rule and the chains rule

Homework Statement How do I know for sure when to use the power rule instead of the chain rule and vice versa? Homework Equations The Attempt at a Solution
4. Integrating a complex ln

Homework Statement integrate:13((4^x)+(3^x))dx Homework Equations The Attempt at a Solution I know the solution is 13((4^x)/ln(4) + (3^x)/ln(3)) + C Can someone explain to me how this works? I don't know where the ln's are coming from. How would I differentiate this back to...
5. Help with a derivative

Homework Statement derivative of: (14x^2)/sqrt(1-x) Homework Equations The Attempt at a Solution I am at this point at finding the derivative: (28x(sqrt(1-x))+((7x^2)/sqrt(1-x))/(1-x) I am confused about the next step because don't know how it's derived from the previous...
6. 4th derivative of cos(2x)

Homework Statement see title. Homework Equations no The Attempt at a Solution Ok so the solution is 16cos(2x) but I'm not sure how it is derived to that. I've tried the product rule but it's not working for me. What rule or rules do I use to get this solution?
7. A couple of integration problems

Homework Statement a) S(4 is higher limit, 0 is lower limit) (x^4 - x^2 + 1)dx b) S(pi is higher limit, -pi is lower limit) (cosx + sinx)dx Homework Equations The S is the integration sign The Attempt at a Solution a) = ((x^5)/5)-((x^3)/3)+x I(4 high, 0 low) =...
8. Integrating e^7x

Homework Statement S e^7x Homework Equations no The Attempt at a Solution Ok so I am using U-substitution for this problem but I don't know what to do next. u = 7x, du = 7dx How do I integrate e^u*du?
9. Calculating indefinite integrals

Homework Statement a) S 13(4^x + 3^x)dx b) S (cosx + sec^2x)dx c) S (3-(1/x))dx d) S e^(7x)dx Homework Equations The S is supposed to be the integration sign The Attempt at a Solution Are these correct or at least close? a) = 13((4^x)/(ln(4) + (3^x)/(ln(3))) + C b) =...
10. Taking 2nd, 3rd and 4th derivatives

Homework Statement a) d^2/(dx^2): ln(x+1) b) d^3/(dx^3): x^7 + 4x^6 - x^2 c) d^2/(dx^2): 1/(x + 1) d) d^4/(dx^4): cos(2x) Homework Equations none The Attempt at a Solution Can someone tell me if these are right? a)= -(1/(x + 1)^2) b)=210x^4 + 480x^3 c)= (2/(x + 1)^3) d)=...
11. A quick derivative question

Homework Statement What is the derivative of 14x^2? Homework Equations The Attempt at a Solution Do i bring the 2 in front to get 28x? Or do I use the chain rule?
12. Limits to infinity with natural logs

Homework Statement I'm having trouble calculating these limits. Homework Equations none The Attempt at a Solution 1. lim.......((lnx)^5)/(x)= x->infinity (((lnx)^5)/x)/(x/x)= (((lnx)^5)/x)/1= How do I calculate from here? 2.lim .......(14lnx^2)/(6lnx^3)=...
13. Calculating limits to infinity

Homework Statement Can someone tell if if these look right? Homework Equations none The Attempt at a Solution 1. lim (lnx)^5/x = x->infinity 5lnx/x = (5lnx/x)/(x/x)= (5lnx/x)/1 = 0/1 = 0 2. lim sinx/x^2= x->infinity (sinx/x^2)/(x^2/x^2)=...
14. Limit of ln as x goes to infinity

Homework Statement lim (lnx)^2/x x-->infinity Homework Equations none The Attempt at a Solution =5lnx/x * (1/lnx)/(1/lnx) =5/(x/lnx) How do I calculate x/lnx?
15. Need help with conic intersection algebra

Homework Statement Find intersection points of the following.(Conics are centered to origin) Circle = x^2+y^2=4, Ellipse = (x^2/4)+(y^2/9) = 1 The Attempt at a Solution So far I have this. (BTW I know the solutions are (-2,0) and (2,0) but I'm still unsure how to get there step...
16. Help with continuity of functions

Homework Statement For each of the following functions, find a value of a, (if such a value exists), which makes the function continuous. a) f(x) = {ax^2......x > 3 ............{x - 7.......x ≤ 3 b) f(x) = {sin(ax)....x < (pi) ............{1............x ≥ (pi) c) f(x) = {x^2 +...
17. How to define this function

Homework Statement If f(x) < g(x) < h(x) for all x E R, and if f and h are continuous functions, must g also be continuous? If so, why? If not, can you come up with a counter example? What do you think?
18. Help with a limit equation involving tangent

Homework Statement How do I solve this equation? lim x>0 (x^3 - 2x^2 + x)/tanx I don't know what to do here, please help. Thank you
19. Help with conic intersections using algebra

Homework Statement Find the intersection points of: Line through (3,1) with slope -1/2, and ellipse through (-3,0) and (-2,2). No graphing just algebra. The Attempt at a Solution So far I have b=10/4 and a^2=9. I'm not sure how to finish this problem. Homework Statement Find the...
20. Help with conic intersections

Homework Statement Hello all, I am searching for help with conic intersections and i have a question I would like to ask. -Consider the hyperbola x^2-y^2 = 1. A line in R^2 (All real numbers squared) can intersect this hyperbola in one of four ways: not at all, at one point crossing the...
21. How to determine change in area of matrices tranformations.

Homework Statement Determine what the tranformation does to the square with vertices (0,0), (0,1), (1,1) and (1,0). Draw the image of the square under these tranformations. Then find the change in area of the square under these transformations. a) [1 1] [1 2] b) [0 -1] [2...