Properties of Definite Integrals

If $\displaystyle\int_{1}^{3}$ f(x)dx = 4, $\displaystyle\int_{3}^{5}$ f(x)dx = 7, and $\displaystyle\int_{5}^{8}$ f(x)dx = -2, what is the value of $\displaystyle\int_{1}^{8}$ f(x)dx?

Which of the following properties of definite integrals is illustrated by the statement that the integral of a function over the interval $[a, b]$ is equal to the integral over $[a, c]$ minus the integral over $[b, c]$?

Which property of definite integrals states that the integral of a function over the interval $[a, b]$ is the sum of the function's integrals over the subintervals $[a, c]$ and $[c, b]$?

Which property of definite integrals states that the integral of a function from $a$ to $c$ is equal to the sum of the integral from $a$ to $b$ and the integral from $b$ to $c$?

If $\displaystyle\int_{1}^{4}$ f(x) dx = 7 and $\displaystyle\int_{1}^{6}$ f(x) dx = 15, what is the value of $\displaystyle\int_{4}^{6}$ f(x) dx?

Which property of definite integrals states that the integral of a function over an interval $[a, b]$ is equal to the sum of its integrals over the subintervals $[a, c]$ and $[c, b]$?

Evaluate the definite integral of the piecewise function $$f(x)=\begin{cases} 2*x+1 & 0\le x<1 \\ 3*x-1 & 1\le x\le 3 \end{cases}$$ over the interval [0, 3].

Let $f$ be the piecewise function defined by

$$ f(x) = \begin{cases}
$$2x + 3, & \text{if } 0 \leq x < 2 \\$$
$$4, & \text{if } 2 \leq x \leq 4 $$
\end{cases} $$

What is the value of $$\int_{0}^{4} f(x) dx$$?

The function f is even and continuous on the closed interval [-4, 4]. If $\displaystyle\int_{0}$^4 f(x)dx = 6, what is the value of $\displaystyle\int_{-4}$^4 f(x)dx?

If g is an even function and $\displaystyle\int_{1}$^5 g(x)dx = 8, what is the value of $\displaystyle\int_{-5}$^₋₁ g(x)dx?

Let $$v$$ be a function such that $$v(-x) = -v(x)$$ for all $$x$$. If $$\displaystyle\int_{2}^{5} v(x) dx = 8$$, what is the value of $$\displaystyle\int_{-5}^{-2} v(x) dx$$?

What is the value of the definite integral $$\displaystyle\int_{-1}^{1} \left(x^5 - 2x^3 + x\right) dx$$?

Which property of definite integrals states that the integral of a function over a zero-width interval, such as $[a, a]$, is zero?

All of the following statements about finding the definite integral of a piecewise function are true except:

Let $$t$$ be a function that satisfies the property $$t(-x) = -t(x)$$ for all $$x$$. If $$\displaystyle\int_{1}^{3} (t(x) + x^2) dx = 18$$, what is the value of $$\displaystyle\int_{-3}^{-1} (t(x) + x^2) dx$$?

Which property of integrals allows an improper integral to be expressed as a sum of integrals over adjacent subintervals?

Let g be the function defined by g(x) = {x² for x < 1, 3 for x = 1, 2x for x > 1}. Which of the following statements is true?

Let `h` be a function defined by `h(x) = sin(x)/x` for `x ≠ 0` and `h(0) = 1`. Which of the following expressions is equivalent to `$\displaystyle\int_{-}$π^π h(x) dx`?

Let $$r$$ be a function such that $$r(x) = -r(-x)$$ for all $$x$$. If $$\displaystyle\int_{1}^{3} r(x) dx = 9$$, then $$\displaystyle\int_{-3}^{-1} r(x) dx =$$

Let $$s$$ be a function satisfying $$s(-x) = -s(x)$$ for all $$x$$. If $$\displaystyle\int_{-5}^{5} [s(x)]^2 dx = 40$$, then $$\displaystyle\int_{0}^{5} [s(x)]^2 dx =$$

Let $$q$$ be a function such that $$q(-x) = -q(x)$$ for all $$x$$. If $$\displaystyle\int_{0}^{3} |q(x)| dx = 12$$, what is the value of $$\displaystyle\int_{-3}^{3} |q(x)| dx$$?

Let $$g$$ be a function such that $$g(-x) = g(x)$$ for all $$x$$. If $$\displaystyle\int_{0}^{3} g(x) dx = 7$$, what is the value of $$\displaystyle\int_{-3}^{3} (g(x) - 4) dx$$?

Let $$f$$ be a function such that $$f(-x) = -f(x)$$ for all $$x$$. If $$\int_{0}^{2} f(x) dx = 5$$, what is the value of $$\int_{-2}^{2} (f(x) + 6) dx$$?

Let $$w$$ be a function such that $$w(x) + w(-x) = 8$$ for all $$x$$. If $$\displaystyle\int_{0}^{3} w(x) dx = 21$$, what is the value of $$\displaystyle\int_{-3}^{3} w(x) dx$$?

Which property of definite integrals establishes that the value of an integral over an interval remains the same if the integrand is modified at a finite number of points within that interval?

Let $$f$$ be the function defined by $$f(x) = \begin{cases} x^2 & \text{for } x < 0 \\ -1 & \text{for } x = 0 \\ x & \text{for } x > 0 \end{cases}$$. What is the value of $$\displaystyle \int_{-1}^{1} f(x) dx$$?

Let $$p$$ be a function that satisfies $$p(-x) = p(x)$$ for all $$x$$. If $$\displaystyle\int_{0}^{4} p(x) dx = 15$$ and $$\displaystyle\int_{0}^{4} x \cdot p(x) dx = 24$$, what is the value of $$\displaystyle\int_{-4}^{4} x \cdot p(x) dx$$?

Let $$k$$ be a function such that $$k(x) - k(-x) = 6x$$ for all $$x$$. Given that $$\displaystyle\int_{0}^{2} k(x) dx = 10$$, what is the value of $$\displaystyle\int_{-2}^{2} k(x) dx$$?

All of the following statements about the properties of definite integrals are true EXCEPT:

Let $$\displaystyle\int_{-2}^{1} u(x) dx = 7$$, $$\displaystyle\int_{1}^{4} u(x) dx = -3$$, and $$\displaystyle\int_{-2}^{4} v(x) dx = 8$$. Based on the information provided, which of the following definite integrals cannot be evaluated?

If $$\displaystyle\int_{-3}^{2} w(x) dx = -6$$ and $$\displaystyle\int_{-3}^{2} z(x) dx = 14$$, which of the following statements is true?

If $$\displaystyle\int_{0}^{6} r(x) dx = 15$$ and $$\displaystyle\int_{0}^{6} s(x) dx = -9$$, what is the value of $$\displaystyle\int_{6}^{0} \left(4r(x) + 3s(x)\right) dx$$?

Given that $$\displaystyle\int_{3}^{7} m(x) dx = 20$$ and $$\displaystyle\int_{3}^{7} n(x) dx = -8$$, which of the following is equal to $$-44$$?

Given that $$\displaystyle \int_{1}^{4} f(x) dx = 8$$ and $$\displaystyle \int_{1}^{4} g(x) dx = -2$$, which of the following values cannot be determined from the information provided?

Given that $$\displaystyle\int_{-4}^{0} e(x) dx = 18$$, $$\displaystyle\int_{0}^{6} e(x) dx = -12$$, and $$\displaystyle\int_{-4}^{6} f(x) dx = 30$$, which of the following expressions cannot be evaluated using the information provided?

Let $$f$$ be a continuous function. If $$\displaystyle \int_{0}^{17} f(x) dx = 8$$, $$\displaystyle \int_{17}^{20} f(x) dx = -3$$, and $$\displaystyle \int_{13}^{20} f(x) dx = 7$$, what is the value of $$\displaystyle \int_{0}^{13} f(x) dx$$?

If $$\displaystyle \int_{4}^{-10} g(x) dx = -3$$ and $$\displaystyle \int_{4}^{6} g(x) dx = 5$$, what is the value of $$\displaystyle \int_{-10}^{6} g(x) dx$$?

Given that $$\displaystyle \int_{0}^{1} f(x) dx = 2$$ and $$\displaystyle \int_{0}^{4} f(x) dx = -3$$, what is the value of $$\displaystyle \int_{1}^{4} (3f(x) + 2) dx$$?

Given the values $$\int_{1}^{4} f(x) dx = 8$$ and $$\int_{1}^{4} g(x) dx = -2$$, which of the following cannot be determined?

Given that $$\displaystyle\int_{0}^{5} a(x) dx = 12$$, $$\displaystyle\int_{5}^{9} a(x) dx = -7$$, and $$\displaystyle\int_{0}^{9} b(x) dx = 15$$, which of the following definite integrals cannot be evaluated using the information provided?

If $$\displaystyle\int_{2}^{5} h(x) dx = 12$$ and $$\displaystyle\int_{2}^{5} k(x) dx = 6$$, which of the following expressions can be evaluated?

Given that $$\displaystyle\int_{-1}^{3} p(x) dx = -4$$ and $$\displaystyle\int_{-1}^{3} q(x) dx = 10$$, which of the following values can be determined?

Given that $$\displaystyle\int_{1}^{8} c(x) dx = 24$$, $$\displaystyle\int_{1}^{3} c(x) dx = 10$$, and $$\displaystyle\int_{1}^{8} d(x) dx = -16$$, what is the value of $$\displaystyle\int_{3}^{8} \left(2c(x) - d(x)\right) dx$$?

Let $$f$$ and $$g$$ be continuous functions. If $$\displaystyle \int_{2}^{6} f(x) dx = 5$$ and $$\displaystyle \int_{6}^{2} g(x) dx = 7$$, then $$\displaystyle \int_{2}^{6} (3f(x) + g(x)) dx =$$

Given that $\displaystyle\int_{0}^{2}$ h(t)dt = 8 and $\displaystyle\int_{0}^{2}$ j(t)dt = 5, what is the value of $\displaystyle\int_{0}^{2}$ (2h(t) - 3j(t)) dt?

If f is a continuous function such that $\displaystyle\int_{0}^{4}$ f(x)dx = 7 and $\displaystyle\int_{0}^{2}$ f(x)dx = 3, which of the following statements must be true?

The function p is defined by p(x) = {x²-4x+3 if x ≤ 2, x+1 if x > 2}. If $\displaystyle\int_{0}^{4}$ p(x) dx = K, then $\displaystyle\int_{0}^{4}$ [p(x) + 3] dx =

If $\displaystyle\int_{1}^{4}$ g(x)dx = 7 and $\displaystyle\int_{4}^{6}$ g(x)dx = 3, which of the following statements must be true?

Which of the following is a property of definite integrals?

Suppose $p$ is a function for which $$\int_{0}^{4} p(x)dx = 6$$ and $$\int_{2}^{6} p(x)dx = 10$$. Which of the following statements must be true?

The function F has jump discontinuities at x = 1 and x = 3. If $\displaystyle\int_{0}^{4}$ F(x) dx = 7, which of the following statements must be true?

Which of the following statements is true about the definite integral of a function on an interval?

The property stating that the definite integral of a function from $a$ to $b$ is the negative of the integral from $b$ to $a$ is a direct consequence of which of the following?

Which of the following is a property of the graph of an odd function?