8. Definite Integrals

Evaluating integrals Evaluate the following integrals.

∫₀¹ √𝓍 (√𝓍 + 1) d𝓍

Evaluating integrals Evaluate the following integrals.

∫π/₆^π/³ (sec² t + csc² t) dt

Evaluating integrals Evaluate the following integrals.

∫₁⁴ ((√v + v) / v ) dv

Evaluating integrals Evaluate the following integrals.

∫₋₅⁵ ω³ /√(ω⁵⁰ + ω²⁰ + 1) dω (Hint: Use symmetry . )

Evaluating integrals Evaluate the following integrals.

∫₋π/₂^π/² (cos 2𝓍 + cos 𝓍 sin 𝓍 ― 3 sin 𝓍⁵) d𝓍

Evaluating integrals Evaluate the following integrals.

∫₀⁵ |2𝓍―8|d𝓍

Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.                                                                                                                         

 ∫₀¹ 2e²ˣ d𝓍

Function defined by an integral Let H (𝓍) = ∫₀ˣ √(4 ― t²) dt, for ― 2 ≤ 𝓍 ≤ 2.

(a) Evaluate H (0) .

Function defined by an integral Let H (𝓍) = ∫₀ˣ √(4 ― t²) dt, for ― 2 ≤ 𝓍 ≤ 2.

(e) Find the value of s such that H (𝓍) = sH(―𝓍)

118. Two worthy integrals

b. Let f be any positive continuous function on the interval [0, π/2]. Evaluate

∫ from 0 to π/2 of [f(cos x) / (f(cos x) + f(sin x))] dx.

(Hint: Use the identity cos(π/2 − x) = sin x.)

(Source: Mathematics Magazine 81, 2, Apr 2008)

125. Wallis products Complete the following steps to prove a well-known formula discovered by the 17th-century English mathematician John Wallis.

a. Use a reduction formula to show that ∫ from 0 to π of (sin^m x) dx = (m − 1)/m × ∫ from 0 to π of (sin^(m−2) x) dx, for any integer m ≥ 2.

Displacement from velocity The following functions describe the velocity of a car (in mi/hr) moving along a straight highway for a 3-hr interval. In each case, find the function that gives the displacement of the car over the interval [0,t], where 0 ≤ t ≤ 3.

v(t) = { 30 if 0 ≤ t ≤ 2

50 if 2 < t < 2.5

44 if 2.5 < t ≤ 3

Area by geometry Use geometry to evaluate the following integrals.

∫⁴₋₆ √(24 ― 2𝓍 ― 𝓍²) d𝓍

108. Arc length Find the length of the curve y = ln(x) on the interval [1, e^2].

114. {Use of Tech} Arc length of the natural logarithm Consider the curve y = ln(x).

a. Find the length of the curve from x = 1 to x = a and call it L(a).

(Hint: The change of variables u = sqrt(x^2 + 1) allows evaluation by partial fractions.)