13.02 Antidiffferentiation and exponential functions
Find the primitive function of the following:
f \rq (x) = e^{ 2 x}
f \rq (x) = e^{ - 5 x }
f \rq (x) = e^{ 0.25 x}
f \rq (x) = 9 e^{ 3 x}
f \rq (x) = e^{x} + e^{ - 3 x }
f \rq (x) = e^{ 3 x} - e^{ - 8 x }
f \rq (x) = \dfrac{1}{6} \left(e^{ 0.5 x} + e^{ 3 x}\right)
f \rq (x) = e^{ 2 x} + \sqrt{x}
Find the following indefinite integrals:
\int e^{ - x } dx
\int e^{ 2 x} dx
\int e^{ \frac{1}{2} x} dx
\int 4 e^{ 2 x} dx
\int e^{4 - 3 x} dx
\int e^{3 - 2 x} dx
\int 4 e^{1 - 2 x} dx
\int \left(e^{ 0.5 x} + e^{ 3 x}\right) dx
\int \left(e^{t} - 5\right) dt
\int \left(e^{ 4 t} + t^{2}\right) dt
\int \left(e^{ 3 v} + v^{4}\right) dv
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.
A curve has gradient function \dfrac{d y}{d x} = e^{ 3 x}. Find the equation of the curve if it passes through the point \left( 0, -1 \right).
Consider gradient function \dfrac{d y}{d x} = 8 e^{ 4 x} - 5.
Find \int (8 e^{ 4 x} - 5)dx.
If y = 7 when x = 0, find y in terms of x.
Consider f \rq \left( x \right) = e^{ - 2 x } + 6 \sqrt{x}.
Find \int \left(e^{ - 2 x } + 6 \sqrt{x}\right)dx.
If f \left( 1 \right) = 3, find f \left( x \right).
Consider f' \left( x \right) = \dfrac{3 e^{ 2 x} + 1}{e^{x}}.
Find \int \left(\dfrac{3 e^{ 2 x} + 1}{e^{x}}\right)dx.
If f \left( 0 \right) = 4, find f \left( x \right).
Find the equation of the curve given the gradient function and a point on the curve:
\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).
A curve has gradient function \dfrac{d y}{d x} = e^{ k x} for some constant k. The point \left(1, e^{3}\right) lies both on the gradient function and also the original curve y.
Determine the value of k.
Find y in terms of x.
Find y in terms of x given the following information:
\dfrac{d y}{d x} = k e^{x} + 2, for some constant k
When x = 0: \, y = - 1, and \dfrac{d y}{d x} = 5