1. evaluate ∫ 8sec^3(2x)dx.

Perform the substitution u=

∫ 8sec3(2x)dx=    ?     +c

2. evaluate ∫ sqrt(e^8x-36)dx

Perform the substitution u=

∫ sqrt(e^8x-36)dx=   ?      +c

3. evaluate ∫ e^x / (16-e^2x)dx

Perform the substitution u=

∫ e^x / (16-e^2x)dx = ?     +c

4. evaluate ∫cos^4(7x)dx.

Perform the substitution u=

∫cos^4(7x)dx=   ?   +c

(1 point) Two chemicals A and B are combined to form a chemical C. The rate of the reaction is proportional to the product of the instantaneous amounts of A and B not converted to chemical C. Initially there are 38 grams of A and 14 grams of B, and for each gram of B, 1.2 grams of A is used. It has been observed that 13 grams of C is formed in 5 minutes. How much is formed in 30 minutes? What is the limiting amount of C after a long time ? grams of C are formed in 30 minutes grams is the limiting amount of C after a long time.

Solve the following first order differential equations:

(a) 2/?^ ???/?? = 4?^2? ; ?(0) = −1/ 3

(b) ??/?? + ? = ? ; ?(0) = 5

(c) ??/?? + ? /? = ? 3 ; ? ( 1/2 ) = 1

Find the arclength of y=3x^3/2 on 1≤x≤2

Use induction to solve the problem. Can you show me the steps too? I don't understand how to solve this.

3+4+5+...+(n+2)=1/2n(n+5)

1+5+5²+...+5^(n-1)=1/4(5ⁿ-1)

14.  If the cylinder of largest possible volume is inscribed in a given sphere, determine the ratio of the   volume of the sphere to that of the cylinder.

15.  Determine the first quadrant point on the curve  y²x = 18 which is closest to the point  (2, 0).     

16.  Two cars are traveling along perpendicular roads, car A at 40 mph, car B at 60 mph.  At noon when   car A reaches the intersection, car B is 90 miles away, and moving toward it.  At 1PM, what is   the rate, in miles per hour, at which the distance between the cars is changing?

Solve the given non-homogeneous recurrence relations:

a_n = a_n-1 + 6a_n-2 + f(n)

a)

a_n = a_n-1 + 6a_n-2 - 2ⁿ⁺¹ with a₀ = -4, a₁= 5

b)

a_n = a_n-1 + 6a_n-2 + 5 x 3ⁿ with a₀ = 2, a₁ = 5

c)

a_n = a_n-1 + 6a_n-2 - 36n with a₀ = 10, a₁= 40

You are flying from Joint Base Lewis-McChord JBLM to an undisclosed location 24km south and 212km east. Mt. Rainier is located approximately 56km east and 40km south of JBLM. If you are flying at a constant speed of 800km/hr, how long after you depart JBLM will you be closest to Mt. Rainier?

A tug of war has teachers and the principal challenging the students! In the first round 3 teachers and 5 students came to a draw. In the second round the principal joined the students. The principal and 10 students came to a draw against 8 teachers in the second round. Assuming each teacher has the same strength, and each student has the same strength, who will win the tie breaking round if it is: 15 students and the principal challenging 10 teachers? Show your thinking.

The volume created by revolving the area created from 5x-5, x^.5-1 and x=5 around x=5. Please show both shell and washer method. Answer for shell and washer should be the same.

Determine if the following series converge or diverge. If it converges, find the sum.

a. ∑n=(3^n+1)/(2n) (upper limit of sigma∞, lower limit is n=0)

b.∑n=(cosnπ)/(2) (upper limit of sigma∞ , lower limit is n= 1

c.∑n=(40n)/(2n−1)^2(2n+1)^2 (upper limit of sigma ∞ lower limit is n= 1

d.)∑n = 2/(10)^n (upper limit of sigma ∞ , lower limit of sigma n= 10)

0. Let?:?2(R)⟶?1(R)bedefinedby?(?+?x+?x²)=(?+?)+(?−?)x,where ?, ?, ? are arbitrary constants.
   a. DeterminethetransformationmatrixforT.(6pts)
   b. Find the basis and the dimension of the Kernel of T. (10pts)

   c. Find the basis and the dimension of the Range of T. (10pts) d. Determine if T is one-to-one. (7pts)
   e. DetermineifTisonto.(7pts)

(a) Show that the lines
r 1 (t) = (2,1,−4) + t(−1,1,−1) and r 2 (s) = (1,0,0) + s(0,1,−2)
are skew.
(b) The two lines in (a) lie in parallel planes. Find equations for these two planes. Express your
answer in the form ax+by+cz +d = 0. [Hint: The two planes will share a normal vector n. How would one find n?]
would one find n?]

In an effort to replace the destruction of trees, five regions of the US have been tracking the annual rate of trees being planted (per thousand) in each region and the annual rate of trees being destroyed (per thousand). Additionally, the tree population (in thousands) by region has been recorded for five different years. The data is given in the tables below. Determine the number of trees planted and the number destroyed in each of the years listed.