Evaluate the given integral by using appropriate method in

Question

1. Find dy/dx.

a) y = cosh(2x⁴-3/x+1)

b) y = ln(sinh^-1x)

c) y = cosh(1/x)

2. Evaluate the given integral by using appropriate method in integration.

a) ∫(2x² - 3/x² - 3x + 2)dx

b) ∫(2x² - 10x + 4/(x+2)²(x²+3))dx

c) ∫x^3/2 ln(x²)dx

d) ∫cos(ln x)dx

e) ∫x²e^-3x dx

f) _-1∫²dx/(x-1)³

_­g) _-∞∫⁰ (e^xdx/4+3e^x)

h) ∫√(tan x)sec⁴ x dx

i) ∫(3x³/√(4-x²) dx

j) ∫(√(x²-16)/x) dx

k) ∫(dx/√(4x-x²))

3. Approximate the integral using (a) the trapezoidal approximation T₄ (b) Sampson's approximation S₆

₁∫³(1/3x+1) dx

4. Show that _k=0Σ^∞(x-2)^k = 1/(3-x); if 4 < x < 6.

5. Show that the given sequence is strictly increasing or strictly decreasing.

a) {n/3n+4}^+∞_n=1

b) {ne^-2n}^+∞_n=1

c) {5n/n!}^+∞_n=1

6. Determine whether the series converges.

a) _k=1Σ^∞3/5^k-1

b) _k=2Σ^∞π^k+2

7. Determine whether the sequence converges; if so find its limit.

{(n+1)(n+3)/3n²}^+∞_n=1

8. Find the general term of the sequence.

1, -1/4, 1/27, -1/256