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(15) Evaluate each of the following integrals. (a) Z cos(x) ln(sin(x)) dx (b) Z x arcsin(x...

(15) Evaluate each of the following integrals. (a) Z cos(x) ln(sin(x)) dx (b) Z x arcsin(x 2 ) dx (c) Z 1 0 ln(1 + x 2 ) dx (d) Z 1/4 0 arcsin(2x) dx

(16) Use the table of integrals to evaluate the integrals, if needed. You may need to transform the integrand first.

(a) Z cos(4t) cos(5t)dt

(b) Z 1 cos3 (x) dx

(c) Z 1 x 2 + 6x + 9 dx

(d) Z 1 50 − 2y 2 dy

(e) Z x 3 cos(x 2 )dx (

f) Z x sin(3x 2 ) dx (g) Z dx √ 16x 2 + 36 (h) Z dx √ 81 − 25x 2

(17) Given the region bounded by the curves x + y = 3, 2x + y = 6, and x = 0. Find the volume of the solid generated by revolving this region about the y-axis.

(18) Given the region bounded by the curves x = |y| and x = 2 − y 2 . (a) Find the volume generated by revolving the region about the y-axis.

Expert Solution

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Solution : ( 16 )

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milcah answered 1 year ago

Evaluate the following integrals using trigonometric identities (a) intergal sin6 x cos3 x dx (b) Z...

Evaluate the following integrals using trigonometric identities (a) intergal sin6 x cos3 x dx (b) Z π/2 o cos5 x dx (c) Z sin3 ( √ x) √ x dx (d) Z tsin2 t dt (e) Z tan2 θ sec4 θ dθ (f) Z x sec x tan x dx

1) Find the following indefinite integrals. a) (4-3xsec^2 x)/x dx b) (5 sin^ 3...

1) Find the following indefinite integrals. a) (4-3xsec^2 x)/x dx b) (5 sin^ 3 x ) / (1+cosx)(1-cosx) dx 2) A particle starts from rest and moves along the x-axis from the origin at t = 0 with acceleration a(t) = 6 - 2t (ms^-2) at time t. When and where will it come to rest. Remember dvdt = acceleration and dsdt = velocity 3) Use substitution to find the following integrals. a) (9x)/ sqrt...

Use integration by parts to evaluate the integral: ∫ cos (ln(6x)) dx

evaluate C (y + 4 sin x) dx + (z2 + 8 cos y) dy +...

evaluate C (y + 4 sin x) dx + (z2 + 8 cos y) dy + x3dz where C is the curve r(t) = sin t, cos t, sin 2t , 0 ≤ t ≤ 2π. (Hint: Observe that C lies on the surface z = 2xy.)

2. Evaluate the following limits. (a) lim x→1+ ln(x) /ln(x − 1) (b) limx→2π x sin(x)...

2. Evaluate the following limits. (a) lim x→1+ ln(x) /ln(x − 1) (b) limx→2π x sin(x) + x ^2 − 4π^2/x − 2π (c) limx→0 sin^2 (3x)/x^2

Find dy/dx for a & b a) sin x+cos y=1 b) cos x^2 = xe^y c)Let...

Find dy/dx for a & b a) sin x+cos y=1 b) cos x^2 = xe^y c)Let f(x) = 5 /2 x^2 − e^x . Find the value of x for which the second derivative f'' (x) equals zero. d) For what value of the constant c is the function f continuous on (−∞,∞)? f(x) = {cx^2 + 2x, x < 2 , 2x + 4, x ≥ 2}

Problem 7. Consider the line integral Z C y sin x dx − cos x dy....

Problem 7. Consider the line integral Z C y sin x dx − cos x dy. (Please show all work) a. Evaluate the line integral, assuming C is the line segment from (0, 1) to (π, −1). b. Show that the vector field F = is conservative, and find a potential function V (x, y). c. Evaluate the line integral where C is any path from (π, −1) to (0, 1).

Question

(15) Evaluate each of the following integrals. (a) Z cos(x) ln(sin(x)) dx (b) Z x arcsin(x...

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