Antiderivatives

A function $F$ satisfies $F'(x) = 6x^{2} - 4x + 1$ and $F(0) = 2$. Find $F(1)$.

Which of the following is the most general antiderivative of $f(x) = 3\sin x - 2$?

A function $F$ is defined for $x < 0$ and satisfies $F'(x) = \dfrac{1}{x}$ there. Which of the following gives every such $F$?

Find the function $f$ with $f'(x) = 4\cos(2x) + e^{x}$ and $f(0) = 1$, and evaluate $f\!\left(\dfrac{\pi}{2}\right)$.

A function $F$ satisfies $F'(x) = \dfrac{3}{1 + 9x^{2}}$ and $F(0) = 0$. Find $F\!\left(\dfrac{1}{3}\right)$.

A ball is thrown straight up from a height of 1 m with an initial velocity of 14.7 m/s. Its acceleration is a constant $-9.8$ m/s². Find the maximum height it reaches.

A colony of bacteria grows at a rate of $200e^{0.1t}$ individuals per hour, where $t$ is measured in hours. By how much does the colony grow during the first 5 hours?

A function $F$ satisfies $F'(x) = 6x^{2}e^{x^{3}}$ and $F(0) = 0$. Find $F(1)$.

A function $f$ satisfies $f''(x) = 12x - 6$, $f'(1) = 2$ and $f(0) = 5$. Find $f(2)$.

A function $f$ is defined for all $x \neq 0$ and satisfies $f'(x) = \dfrac{1}{x}$ for every $x \neq 0$, with $f(-1) = 0$ and $f(1) = 3$. Find $f(-e) + f(e)$.