Direction: Using Riemann's Sum, find the integral of the following functions: 1.) √²(2x² - 5x + 1) dx 2.) √(x³- 2x + 1) dx 3.) f 2x³ dx 2. Find the definite integral ("x" Ax uving Reimann's Sum " a=0 f(xi) Xat (4x) i = 0 + ( ²/√) i 4x-6-4 n b-2 2-0 1 n a ai 4x= xi= S 11 n " f(x) = xn == (xi) = ( 2 1 / 7 f(xi) = л 817 n" Lim 818 n a F * n (n+1) a 1-1 a 8 n(n+1) n 33 Q 8 n4 + ... n n' 4 n AX n" ( = 1 a 817 Vimplify: n n" 14 nyt... = (= | n4 4 a 8 .27 14 = n п 4 пр (=T = 4(1) 4

f(2) = ; f' (x) = -1/x^2 f"(x) = 2/x^3 f" (x) = -6/x^4 f(m) (x) = (1/x)^n Hint: The expression (–1)" has value 1 if n is even and –1 if n is odd. This expression can be used in your answer for the last part.]

Solve the ODE xdx + ydy = xdy - ydx a. ln [(y+x) / x] = C b. ln [(x-y) / x] = C c. arctan (y/x) - 0.5 ln [(y^2 + x^2) / x] = C d. arctan (y/x) - 0.5 ln [(y^2 + x^2) / x^2] - lnx = C e. none

Find the indefinite integral and check the result by differentiation. 1 Step 1 Observe that 7 10x (7- x²) ³ Step 2 Multiply and divide the integrand by Now, g'(x) dx = 10x 7-x² -3 dx = √ (7²0x²) ³ x - B5 / (²-x²) ²³ (-2x) ax (フー×2) -5 dx = f(g(x))g'(x) dx -2x Rewrite the given integral in terms of g(x). S= = - -5 is the derivative of (7- x²). Let g(x) = (7 - x²). =/35 — — Submit Skip (you cannot come back) and rewrite the integral. -3 . Define f(g(x)) = -5(7-x²) -³ such that f(x) = -5 -/0 [F(g( S F(g(x)) + C where F(g(x)) is the antiderivative of f(g(x)). -3 1/ √ √ (7 - x²)-³(-2x) dx (x))g'(x) dx ] ›)] + c C

Where is S(x) = 4(x-2)^ + 2(x-2)* increasing?

For a positive integer n, prove that summation d/n mu^(2)(d) phi(d) = n / phi(n). [Hint: Both sides of the equation are multiplicative functions].

Letf" () 8t 10Ví (a) Find the most general formula for f"(t). If an arbitrary constant must be used here, use an upper-case "C". f"(t)4t^2-20/3t (3/2)+C (b) Based on your answer to (a), find the most general formula for f'(t). If another new arbitrary constant must be used here, use an upper-case "D" f'()= (c) Based on your answer to (b), find the most general formula for f(t). If another new arbitrary constant must be used here, use an upper-case "E" f(t)

2. Prove that dx (2x+1)}] (2x+1) (4x-19) X-3 (x-3)² = Then, -5 (2x+1)²(8x-38) 3(x-3)² -dx.

What is q′(x) when q(x)=ln((x^2+1)(x^3+x^2+2))?

Prove that for integers y=1 the equation 3 h-n (2 x ₁ + x₂ + .. + x₂) +3²(1)(2x₁+x₂+.+xn-₁) +...+ 3^-1 (2²₁) Y = 2x₁+x₂+...+% -3" cannot be solved with positive integers X₁₁X₂... Xny no