If $$\int f(x) dx = x^2\ln(x) - \frac{x^2}{2} + C$$, then $$f(x) =$$
Solving for unknown constants, limits of integration, or the function itself.
29 questions
AP Learning Target(s): LT 7: The Fundamental Theorem of Calculus and Definite Integrals, LT 8: Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation
If $$\int f(x) dx = x^2\ln(x) - \frac{x^2}{2} + C$$, then $$f(x) =$$
$$x^2/x + 2x\ln(x) - x$$
$$x(2\ln(x) - 1)$$
$$2x\ln(x)$$
$$x^2\ln(x) - x$$
If $$\int_{0}^{x} (4t-1) dt = 15$$, what is the value of $$x$$?
$$3$$
$$\frac{11}{2}$$
$$-2.5$$
$$2.5$$
If $$\int f(x) dx = x^2\sin(x) - 2x\cos(x) + 2\sin(x) + C$$, then $$f(x) =$$
$$x^2\cos(x) + 4x\sin(x)$$
$$2x\sin(x) - 2\cos(x)$$
$$x^2\sin(x) + 2x\cos(x)$$
$$x^2\cos(x) + 2x\sin(x)$$
What is the approximate value of x such that $$\int_{1}^{x} (2t^{1/3}-1) dt = 5$$?
$$\approx 3.50$$
$$\approx 5.00$$
$$\approx 3.95$$
$$\approx 4.50$$
What is the value of $$x$$ for which $$\int_{0}^{x} \cos(t) dt = 1$$?
$$\pi$$
$$\frac{\pi}{2}$$
$$\frac{\pi}{4}$$
$$\frac{3\pi}{2}$$
What is the value of $$x$$ for which $$\int_{2}^{x} 2t dt = 10$$?
$$\sqrt{14}$$
$$\sqrt{10}$$
$$\sqrt{24}$$
$$2+\sqrt{10}$$
Find the value of $$x > 0$$ that satisfies the equation $$\int_{0}^{x} (2t+5) dt = \int_{0}^{3} (2t+5) dt$$.
$$8$$
$$-8$$
$$11$$
$$3$$
Find the value of $$x$$ such that $$\int_{x}^{2x} t dt = 18$$.
$$2*\sqrt{3}$$
$$\sqrt{18}$$
$$2*\sqrt{2}$$
$$\sqrt{6}$$
What is the value of $$x$$ for which $$\int_{-1}^{x} (3t+2) dt = 10.5$$?
$$2+\frac{1}{2}$$
$$\frac{20}{3}$$
$$-\frac{10}{3}$$
$$2$$
Find all values of $$x$$ for which $$\int_{4}^{x}(t-2) dt=0$$.
$$0$$ and $$4$$ only
$$4$$ only
$$0$$ only
$$2$$ and $$4$$ only
For which values of $$x$$ is $$\int_{-1}^{x}(t^2-1) dt=0$$?
$$2$$ only
$$-1$$ only
$$-1$$ and $$2$$ only
$$-1$$, $$1$$, and $$2$$ only
What are all values of $$x$$ for which $$\int_{3}^{x}(t^2-9) dt=0$$?
$$-3$$, $$3$$, and $$6$$ only
$$-6$$ and $$3$$ only
$$-6$$ only
$$3$$ only
For what values of $$x$$ is $$\int_{2}^{x}(t^2-4) dt=0$$?
$$-4$$ only
$$-4$$ and $$2$$ only
$$-2$$, $$2$$, and $$4$$ only
$$2$$ only
For which values of $$x$$ is $$\int_{-2}^{x}(t+1) dt=0$$?
$$-1$$, $$0$$, and $$2$$ only
$$0$$ only
$$-2$$ and $$0$$ only
$$-2$$ only
Find all values of $$x$$ for which $$\int_{0}^{x}(2t-6) dt=0$$.
$$6$$ only
$$0$$ only
$$3$$ and $$6$$ only
$$0$$ and $$6$$ only
For what values of $$x$$ is $$\int_{1}^{x}(t^2-1) dt=0$$?
$$1$$ only
$$-1$$, $$1$$, and $$2$$ only
$$-2$$ only
$$-2$$ and $$1$$ only
What are all values of $$x$$ for which $$\int_{-1}^{x}(t^3-t) dt=0$$?
$$-1$$ and $$1$$ only
$$-1$$, $$0$$, and $$1$$ only
$$1$$ only
$$-1$$ only
Find the value of $$x$$ that satisfies the equation $$\int_{1}^{x} \ln(t) dt = 1$$.
$$0$$
$$e$$
$$1$$
$$e-1$$
Find the value of $$x$$ that satisfies the equation $$\int_{1}^{x} \frac{1}{t} dt = \ln(5)$$.
$$\ln(5)$$
$$e^5$$
$$5$$
$$1/5$$
What is the value of the constant $$c$$ for which $$\int_{0}^{2} (ct+3) dt = 10$$?
$$4$$
$$2+\frac{1}{3}$$
$$3$$
$$2$$
If $$\int_{2}^{5} k dt = 21$$, what is the value of $$k$$?
$$7$$
$$21/5$$
$$3$$
$$8$$
What is the value of $$x$$ for which $$\int_{1}^{x} 3t^2 dt = 26$$?
$$3$$
$$27$$
$$1$$
$$4$$
What is the value of $$x$$ if $$\int_{3}^{x} 4 dt = 20$$?
$$8$$
$$10$$
$$12$$
$$5$$
What is the value of $$x$$ for which $$\int_{4}^{x} \sqrt{t-4} dt = \frac{16}{3}$$?
$$7$$
$$4+\sqrt[3]{16}$$
$$\frac{20}{3}$$
$$8$$
Find the value of $$x$$ such that $$\int_{4}^{x} (t-4)^3 dt = 64$$.
$$6$$
$$10$$
$$4+\sqrt{64}$$
$$8$$
What are all values of $$x$$ that satisfy the equation $$\int_{2}^{x}(t-1) dt=0$$?
$$-2$$, $$0$$, and $$2$$ only
$$0$$ only
$$2$$ only
$$0$$ and $$2$$ only
Find the value of $$x$$ for which $$x > 0$$ and $$\int_{0}^{x} (2t+3) dt = 28$$.
$$7$$
$$-4$$
$$-7$$
$$4$$
For what values of $$x$$ is $$\int_{0}^{x}(t^2-1) dt=0$$?
$$0$$ and $$\sqrt{3}$$ only
$$-\sqrt{3}$$, $$0$$, and $$\sqrt{3}$$ only
$$\sqrt{3}$$ only
$$0$$ only
Find the value of $$x$$ that satisfies the equation $$\int_{0}^{x} e^t dt = e^2-1$$.
$$e^2$$
$$2$$
$$1$$
$$\ln(e^2-1)$$