Hey guys, I have a few problems that I'm having trouble with. If anyone could help, I would really appreciate it.

Here are the instructions:

Problem 1.Each of these problems requires an integral that can be calculated either by making a trigonometric substitution or by using a Table of Integrals. In each case, do the following:

i) Write the integral, including the limits of integration, that will produce the required quantity.

ii) Write an antiderivative for your integral. You may do this by direct calculation, or you may use a Table of Integrals.

iii) Complete the calculation to compute the required quantity.

a)Calculate the length along the curve y = ln(x), 1 <= x <= e

My Solution

ai) $\displaystyle y = ln(x), \frac{dy}{dx} = \frac{1}{x}$

$\displaystyle \int^e_1\sqrt{1 + \frac{1}{x^2}}$ dx

aii) Here, I don't know what to do. I can't find any form in my textbook's table of integrals that matches this, nor do I know where to start by using a trigonometric substitution. Help please!

b)Calculate the length along the parabola $\displaystyle y = x^2$ between the points (0,0) and (1,1).

My Solution

bi) $\displaystyle y = x^2, \frac{dy}{dx} = 2x$

$\displaystyle \int^1_0\sqrt{1 + 4x^2}$ dx

bii) Once again, I can't figure out what to do here. I tried substituting $\displaystyle \tan\theta$ for 2x, but it isn't working out right. Help please!

c)Calculate the area bounded by the ellipse $\displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.

My Solution

ci) $\displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, y = \sqrt{b - \frac{bx}{a}}$

$\displaystyle \int^a_{-a}\sqrt{b - \frac{bx}{a}}$ dx

cii) I'm in the same boat here, I have no idea what to do. Help please!

Problem 2.Make a complete calculation of $\displaystyle \int\frac{3x^6 + 27x^4 + 2x^3 + 2x^2 + 45}{x^4 + 9x^2}$ dx.

= $\displaystyle \int\frac{3x^6 + 27x^4 + 2x^3 + 2x^2 + 45}{x^2(x^2 + 9)}$ dx = $\displaystyle \int(\frac{A}{x} + \frac{B}{x^2} + \frac{Cx + D}{x^2 + 9})$

$\displaystyle 3x^6 + 27x^4 + 2x^3 + 2x^2 + 45 = A(x)(x^2 + 9) + B(x^2 + 9) + (Cx + D)(x^2)$ = $\displaystyle (A + C)x^3 + (B + D)x^2 + (9A)x + 9B$

A + C = 2, B + D = 2, 9A = 0, 9B = 45

A = 0, B = 5, C = 2, D = -3

= $\displaystyle \int(\frac{5}{x^2} + \frac{2x - 3}{x^2 + 9})$ dx = $\displaystyle \int\frac{5}{x^2}$ dx + $\displaystyle \int\frac{2x}{x^2 + 9}$ dx - $\displaystyle \int\frac{3}{x^2 + 9}$ dx = $\displaystyle 5ln(x^2) + ln(x^2 + 9) - 3ln(x^2 + 9) + C$

Could someone check to see if I did this problem correctly? Thank you very much in advance.