Find the primitive function of the following:
Find the following indefinite integrals:
Find the exact value of the following definite integrals:
Find the primitive function of the following:
Consider the function y = x^{2} e^{x}.
Find an expression for \dfrac{dy}{dx}.
Hence find \int x \left( 2+ x \right) e^{x} dx.
Find the exact area enclosed by:
The curve y = e^{ 4 x}, the x-axis, and the lines x=1 and x=2.
The curve y = e^{x}, the x-axis, the y-axis, and the line x = 2.
The curve y = -e^{x}, the x-axis, the y-axis, and the line x = 5.
The curve y = e^{x} + 4, the coordinate axes, and the line x = 4.
The curve y = e^{ - 6 x}, the x-axis, and the lines x = - 2 and x = 1.
The curve y = 6 e^{ 3 x}, the x-axis and the lines x = - 3 and x = 3.
The curve y = -2^{ x}, the x-axis, and the lines x=1 and x=2.
Consider the curve y = e^{x} - 9.
Find the exact value of the x-intercept.
Find the exact area between the curve y = e^{x} - 9 and the x-axis, from the lines x = 0 to x = 3.
Consider the curve y = e^{x}.
Find the area bound by the curve, the x-axis, the y-axis, and the line x = 3.
Find the equation of the tangent to the curve y = e^{x} at the point where x = 3.
Find the exact area enclosed between y = e^{x}, the x-axis, the y-axis, and the tangent at x = 3.
Find the equation of the curve given the gradient function and a point on the curve:
\dfrac{d y}{d x} = e^{ 3 x}, and point \left(0, -1\right).
\dfrac{d y}{d x} = e^{ 2 x}, and point \left(0, 6 \right).
\dfrac{d y}{d x} = e^{ x} -2x, and point \left(0, 4 \right).
f' \left( x \right) = {x^2}e^{2x^3}, and point \left(0, 0\right).
A particle P starts off from a fixed point O, with an initial velocity, v, of 2\text{ m/s}. Its acceleration a \text{ m/s}^2 after t seconds is given by:
a = e^{ - t }Find v\left(4 \right), the velocity of P after 4 seconds, correct to two decimal places.
Find the displacement, x, of the particle P after 4 seconds, correct to the nearest hundredth of a metre.
An object is cooling and its rate of change of temperature, after t minutes, is given by: T' = - 10 e^{ - \frac{t}{5} }
Find the instantaneous rate of change of the temperature after 8 minutes, to two decimal places.
Find the total change of temperature after 9 minutes, to two decimal places.
Hence, find the average change in temperature over the first 9 minutes, to two decimal places.
Fluid leaks from a storage facility and its rate of change after t days is given by:
F' \left( t \right) = 3000 e^{ - 0.5 t}Find the amount of fluid that will leak in the first 7 days, correct to two decimal places.
Find the amount of fluid that will leak in the next 7 days, correct to two decimal places.
A particle's acceleration a, is measured in\text{ cm/s}^2 according to the equation:
a = - e^{ 4 t}If the particle is initially at rest, find an equation for the velocity, v, of the particle at time t.
Find the change in the particle's position after 5 seconds, to the nearest centimetre.
The initial velocity of a particle is 10\text{ m/s}, and its acceleration in \text{ m/s}^2 is given by:
a \left( t \right) = 2 e^{ 3 t}Find its velocity, v \left( t \right), after t = 2 seconds. Round your answer to the nearest metre per second.
Let x \left( t \right) be the displacement of the particle at t seconds. If its initial position is 2 \text{ m} to the left of the origin, find its displacement after 4 seconds, correct to the nearest metre.
A particle moves so that its velocity over time t, in metres per second, is given by:
v = 2 e^{t} - 1If the particle's displacement, x, is 10 when t = 0, find its exact displacement when t = 3.
A rechargeable battery has a voltage of 9 \text{ V} when fully charged. When the battery is used to run an electronic toy, the voltage, V volts, remains at 9 \text{ V} for 30 minutes and then it decreases instantaneously, at a rate modelled by:
\dfrac{d V}{d t} = - 0.3 e^{ - 0.03 t}Find the change in the battery voltage after the toy has been in use for 45 minutes. Round your answer to two decimal places.
Find the function V, the battery voltage after the toy has been in use for t minutes when t \geq 30.
Find the number of whole minutes the battery can be used to run this toy, if a minimum voltage of 7 \text{ V} is required.