8. Definite Integrals

Using properties of integrals Use the value of the first integral I to evaluate the two given integrals. 

I = ∫₀^π/2 (cos θ ― 2 sin θ) dθ = ―1

(a) ∫₀^π/2 (2 sin θ ― cos θ) dθ

Evaluate the integrals in Exercises 41–60.

51. ∫(from ln2 to ln4)coth(x)dx

Evaluating Definite Integrals

Evaluate the integrals in Exercises 47–68.

∫₋₁¹ (3x² - 4x + 7)dx

Evaluate the line integral ${\int }_C^{}xy ds$, where C is the curve given by $x=t^2$, $y=2t$, for $0\le t\le 1$.

Zero net area Consider the function ƒ(𝓍) = 𝓍² ― 4𝓍 .                                                                                                                                       

                                                                                                                                                                                     c) In general, for the function ƒ(𝓍) = 𝓍² ― a𝓍, where a > 0, for what value of b > 0 (as a function of a) is ∫₀ᵇ ƒ(𝓍) d𝓍 = 0 ? 

Without evaluating integrals, prove that ∫₀² d/dx(12 sin πx²) dx=∫₀² d/dx (x¹⁰(2−x)³) dx.

Evaluating integrals Evaluate the following integrals.

∫₁⁴ ((√v + v) / v ) dv

Properties of integrals Consider two functions ƒ and g on [1,6] such that ∫₁⁶ƒ(𝓍) d𝓍 = 10 and ∫₁⁶g(𝓍) d𝓍 = 5, ∫₄⁶ƒ(𝓍) d𝓍 = 5 , and ∫₁⁴g(𝓍) d𝓍 = 2. Evaluate the following integrals.

(b) ∫₁⁶ (f(𝓍) ― g(𝓍)) d𝓍

{Use of Tech} Powers of sine and cosine It can be shown that

∫ from 0 to π/2 of sinⁿx dx = ∫ from 0 to π/2 of cosⁿx dx =

{

[1·3·5···(n-1)]/[2·4·6···n] · π/2 if n ≥ 2 is even

[2·4·6···(n-1)]/[3·5···n] if n ≥ 3 is odd

}

b. Evaluate the integrals with n = 10 and confirm the result.

Evaluate the line integral of the vector field $F(x,y)=(y,x)$ along the curve $C$, which is the line segment from $(0,0)$ to $(1,1)$.

2. Evaluate the definite integral from $1$ to $e$: ${\int }_1^{e}xln\left(x\right)dx$.

Given the region bounded by $y=x^2$, $x=0$, and $y=4$, determine by direct integration the x-coordinate of the centroid of the area.

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)

Prove the following orthogonality relations (which are used to generate Fourier series). Assume m and n are integers with m ≠ n.

c.

π

∫ sin(mx) cos(nx) dx = 0, when |m + n| is even

0

Find the area of the region enclosed by one loop of the curve $r=\sin \left(8\theta \right)$.