MAT 302 Assignment Problem Natural Log Functions Integration

MAT 302  Assignment Problem Natural Log Functions Integration | Borough of Manhattan Community College

Question 1

(1 point) Evaluate the indefinite integral.

∫x+2x2+4x+5dx

Question 2

(1 point) Find the following integral. Note that you can check your answer by differentiation.

Question 3

(1 point) Find the following integral. Note that you can check your answer by differentiation.

Question 4

(1 point) Evaluate the indefinite integral.

∫4dxxln(6x)

Question 5

1 point) Evaluate the following definite integral.

∫20167x−9dx

Question 6

(1 point) Logarithms as anti-derivatives.

∫−6x(lnx)2dx

Question 7

(1 point) Evaluate the definite integral.

∫e31dxx(1+lnx)∫1e3dxx(1+ln⁡x)

Question 8

(1 point) Consider the function

f(x)={x1xif x<1if x≥1f(x)={xif x<11xif x≥1

Evaluate the definite integral.

∫4−2f(x)dx

Question 9

(1 point) Find the indefinite integral by uu - substitution.

∫11+2x−−√dx

Question 10

(1 point) Solve the differential equation.

drdt=sec2(t)tan(t)+1drdt=sec2(t)tan(t)+1

Question 11

Part I

Derive the formula for the indefinite integral of csc(x).

∫csc(x)dx∫csc(x)dx

Part 1:

Multiply csc(x)csc(x) by csc(x)+cot(x)csc(x)+cot(x)csc(x)+cot(x)csc(x)+cot(x) to get

(csc2(x)+csc(x)cot(x))/(csc(x)+cot(x))

Part 2:

Let uu be the denominator of the previous answer

dudu = (−csc2(x)−csc(x)cot(x))dx

Part 3:

Rewrite the integral using uu

∫csc(x)dx∫csc(x)dx = ∫∫ (−1/u)du

Part 4:

Integrate your previous answer:

−ln|u|+C

Hint:

Part 5:

Substitute back uu and give the answer

∫csc(x)dx∫csc(x)dx = −ln|csc(x)+cot(x)|+C

Part II

Evaluate the definite integral.

∫π4π8(csc2θ−cot2θ)dθ