8. Definite Integrals

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus

∫₁² 3/t dt

Derivatives of integrals Simplify the following expressions.

d/d𝓍 ∫₀ˣ (√1 + t²) dt (Hint: ∫ˣ₋ₓ (√1 + t²) dt = ∫⁰₋ₓ (√1 + t²) dt + ∫ˣ₋ₓ (√1 + t²) dt ) .

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

(c) Evaluate H '(2) .

Area functions and the Fundamental Theorem Consider the function

ƒ(t) = { t      if  ―2 ≤ t < 0

t²/2    if    0 ≤ t ≤ 2

and its graph shown below. Let F(𝓍) = ∫₋₁ˣ ƒ(t) dt and G(𝓍) = ∫₋₂ˣ ƒ(t) dt.

(c) Use the Fundamental Theorem to find an expression for F '(𝓍) for 0 ≤ 𝓍 < 2.

143.

a. Show that ∫ ln(x) dx = x ln(x) − x + C.

Evaluate the following integral:

${\int }_1^2\frac{5}{x^2}dx$

Derivatives of integrals Simplify the following expressions.

d/d𝓍 ∫₃ˣ (t² + t + 1) dt

Cubic zero net area Consider the graph of the cubic y = 𝓍 (𝓍― a) (𝓍― b), where 0 < a < b. Verify that the graph bounds a region above the 𝓍-axis, for 0 < 𝓍 < a , and bounds a region below the 𝓍-axis, for a < 𝓍 < b. What is the relationship between a and b if the areas of these two regions are equal? 

Evaluate the following integral:

${\int }_0^1(2x^3-x^2+4x)dx$

{Use of Tech} Fresnel integrals The theory of optics gives rise to the two Fresnel integrals

S(x) = ∫₀ˣ sin t² dt and C(x) = ∫₀ˣ cos t² dt

a. Compute S′(x) and C′(x).

In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.

y = ∫(from 0 to lnx) sin(e^t) dt

Evaluate the following integral:

${\int }_2^3{x}^{\frac{5}{2}}dx$

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus

∫¹₁/₂ (t⁻³ ― 8) dt

Derivatives of integrals Simplify the following expressions.

d/dy ∫¹⁰ᵧ³ √(𝓍⁶ + 1) d𝓍

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ and ƒ' are continuous functions for all real numbers.

(b) Given an area function A(𝓍) = ∫ₐˣ ƒ(t) dt and an antiderivative F of ƒ, it follows that A'(𝓍) = F(𝓍) .