Evaluate the integral. (Use C for the constant of integration.)

(x^2-1)/(sqrt(25+x^2)*dx

Evaluate the integral. (Use C for the constant of integration.)

dx/sqrt(9x^2-16)^3

Evaluate the integral. (Use C for the constant of integration.)

3/(x(x+2)(3x-1))*dx

?" + 3?′ + 2? = ????, ?(0) = 0, ?′(0) = 2

1) Please solve using an annihilator

2) Please solve using the Method of Variation of Parameters

Thank you.

A bicycle shop sells two styles of a road bike, 10-speed and 14-speed. During the month of September, the management expects to sell exactly 45 road bikes. The monthly profit is given by P(x,y)=−(1/9)x^2−5y^2−(1/9)xy+10x+65y−100, where x is the number of 10-speed road bikes sold and y is the number of 14-speed road bikes sold. How many of each type should be sold to maximize the profit in September?

Instructions: For each solid described, set up, BUT DO NOT EVALUATE, a single definite integral that represents the exact volume of the solid. You must give explicit functions as your integrands, and specify limits in each case. You do not need to evaluate the resulting integral.

1. The solid generated by rotating the region enclosed by the curves y = x^2 and y = x about the line x-axis.

1) Evaluate the integral from 0 to 1 (e^(2x) (x^2 + 4) dx)
(a) What is the first step of your ‘new’ integral?
(b) What is the final antiderivative step before evaluating?
(c) What is the answer in simplified exact form?

2) indefinite integral (cos^2 2theta) / (cos^2 theta) dtheta
(a) What is the first step of your ‘new’ integral?
(b) What is the simplified integral before taking the antiderivative?
(c) What is the answer in simplified form?

A mass of 1.5 kg stretches a spring 0.05 mm. The mass is in a medium that exerts a viscous resistance of 240 NNwhen the mass has a velocity of 6 msms. The viscous resistance is proportional to the speed of the object.

Suppose the object is displaced an additional 0.06 mm and released.

Find an function to express the object's displacement from the spring's natural position, in mm after tt seconds. Let positive displacements indicate a stretched spring, and use 9.8 ms2ms2 as the acceleration due to gravity.

Find the directional derivative of f at the given point in the direction indicated by the angle θ.

f(x, y) = y cos(xy),    (0, 1),    θ = π/4

D_uf(0, 1) =

Find all distinct roots (real or complex) of z2+(−6+i)z+(25+15i). Enter the roots as a comma-separated list of values of the form a+bi. Use the square root symbol '√' where needed to give an exact value for your answer. z = ???

Given that v⃗ 1=[−3,1] and v⃗ 2=[2,−1] are eigenvectors of the matrix

determine the corresponding eigenvalues. λ1= . λ2= .

Given the curve C in parametric form :

C : x = 2cos t , y = 2sin t , z = 2t ; 0≤ t ≤ 2pi

a) the velocity v(t)

b) the speed ds/dt

c) the acceleration a(t)

d) the unit tangent vector T(t)

e) The curvature k and the normal vector N(t)

f) the binormal vector B(t)

g) The tangential and normal components of accelertation

Questions Determine whether or not the vector field is conservative. If it is conservative, find a vector f f such that . F→=∇f. → F ( x , y , z ) =< y cos x y, x cos x y , − sin z > F→ conservative. A potential function for → F F→ is f ( x , y , z ) = f(x,y,z)= + K. (Type "DNE" if → F F→ is not conservative.)

Use the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.)

dx,    n = 4

(a) the Trapezoidal Rule


(b) the Midpoint Rule


(c) Simpson's Rule

Find all distinct (real or complex) eigenvalues of A. Then find the basic eigenvectors of A corresponding to each eigenvalue.
For each eigenvalue, specify the number of basic eigenvectors corresponding to that eigenvalue, then enter the eigenvalue followed by the basic eigenvectors corresponding to that eigenvalue.

A = 11 −10

17 −15

Number of distinct eigenvalues: ?

Number of Vectors: ?

? : {???}