x for 0 &lt

Faculty of Engineering Mathematical Analysis I

Fall 2018 Exercises 5

Mean Value Theorem, Techniques of Integration 1. Using Mean Value Theorem show that √

1 + x < 1 +^x₂ for x > 0 and for −1 ≤ x < 0.

2. Show that tan x > x for 0 < x < π/2.

3. Suppose that f (0) = −3 and f⁰(x) ≤ 5 for all values of x. How large can f (2) possibly be?

(Hint: Use MVT)

4. Evaluate the following indefinite integrals (a) R sin (3 ln x)

x dx

(b) R e^x√

1 + e^xdx (c) R x³cos x²
dx (d) R e^{arctan x}

1 + x² dx (e) R cos⁶xdx (f) R sin³x cos²xdx (g) R sin⁶x cos³xdx (h) R sin²x cos²xdx (i) R tan³x sec⁷xdx (j) R tan²xdx (k) R tan⁵xdx

(l) R cot³x csc³xdx (m) R cot⁶x csc⁴xdx

(n) R cos x cos⁵(sin x) dx (o) R x²

√9 − x²dx

(p) R 1

(4 + x²)^3/2 dx

(q) R

√x²− 1

x dx

(r) R

√x − 1 + 1

√3

x − 1 dx (s) R x³

(4x²+ 9)^3/2 dx

(t) R x

√3 − 2x − x²dx (u) R e^xcos xdx (v) R x²e^xdx (w) R x²ln xdx

(x) R x arctan xdx (y) R x⁴+ 2x²+ x

x³+ 1 dx (z) R 3x − 1

x²− 2x − 3dx (ξ) R x + 3

x⁴+ 9x²dx

5. Write out the form of the partial fraction decomposition of the functions (a) 2x + 1

(x + 1)³(x²+ 4)² (b) x⁴

(x³+ x)³(x²− x + 3)

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