Consider the function x^{4}.
Calculate \dfrac{d}{dx} \left(x^{4}\right).
Hence find \int 4 x^{3} dx.
Consider the function x^{6}.
Calculate \dfrac{d}{dx} \left(x^{6}\right).
Hence find \int 24 x^{5} dx.
Consider the function x^{5} + x^{4}.
Calculate \dfrac{d}{dx} \left(x^{5} + x^{4}\right).
Find \int \left( 35 x^{4} + 28 x^{3}\right) dx.
Consider the function x^{ - 4 }.
Calculate \dfrac{d}{dx} \left(x^{ - 4 }\right).
Hence find \int 16 x^{ - 5 } dx.
Consider the function x^{8} + x^{6}.
Calculate \dfrac{d}{dx} \left(x^{8} + x^{6}\right).
Find \int \left( 4 x^{7} + 3 x^{5}\right) dx.
Consider the function \sqrt[5]{x^{6}}.
Calculate \dfrac{d}{dx} \left(\sqrt[5]{x^{6}}\right).
Hence find \int 6 \sqrt[5]{x} dx.
Consider the gradient function f \rq \left( x \right) = 2.
Find the general antiderivative to f \rq \left( x \right).
The gradient function f \rq \left( x \right), is a constant function. What type of function is the antiderivative function, f \left( x \right).
If the form of a constant gradient function is f \rq \left( x \right) = m , write the general form of the antiderivative function.
Consider the gradient function f \rq \left( x \right) = - 8 x.
Find the general antiderivative to f \rq \left( x \right).
The gradient function f \rq \left( x \right), is a linear function. What type of function is the antiderivative function, f \left( x \right).
If the form of a linear gradient function is f \rq \left( x \right) = ax , write the general form of the antiderivative function.
Consider the gradient function f \rq \left( x \right) = 4 x + 3.
Find the general antiderivative to f \rq \left( x \right).
The gradient function f \rq \left( x \right), is a linear function. What type of function is the antiderivative function, f \left( x \right).
If the form of a linear gradient function is f \rq \left( x \right) = ax +b , write the general form of the antiderivative function.
Consider the gradient function f \rq \left( x \right) = 7 x^{2}.
Find the general antiderivative to f \rq \left( x \right).
The gradient function f \rq \left( x \right), is a quadratic function. What type of function is the antiderivative function, f \left( x \right).
If the form of a quadratic gradient function is f \rq \left( x \right) = ax^2 , write the general form of the antiderivative function.
Find the general antiderivative of the following:
\dfrac{d y}{d x} = 9
\dfrac{d y}{d x} = 8 x
\dfrac{d y}{d x} = 10 x + 7
\dfrac{d y}{d x} = 9 x^{2}
\dfrac{d y}{d x} = 9 x^{2} + 4 x - 6
\dfrac{d y}{d x} = x^{5}
\dfrac{d y}{d x} = 15 x^{4} + 16 x^{3}
\dfrac{d y}{d x} = \dfrac{x^{6}}{4} + \dfrac{x^{2}}{3}
\dfrac{d y}{d x} = x^{ - 6 }
\dfrac{d y}{d x} = \dfrac{15}{x^{6}}
\dfrac{d y}{d x} = \dfrac{10}{x^{6}} - \dfrac{9}{x^{4}}
\dfrac{d y}{d x} = 4 x^{\frac{2}{5}} + 3 x^{\frac{4}{7}}
\dfrac{d y}{d x} = x^{ - \frac{3}{7} } + x^{-\frac{2}{5} }
\dfrac{d y}{d x} = 8 x^{3} + 3 x^{\frac{5}{3}} - 3
\dfrac{d y}{d x} = \dfrac{x^{3} + 4}{x^{3}}
\dfrac{d y}{d x} = x^{2} \left( 10 x^{2} - 9 x\right)
\dfrac{dy}{dx} = \left( 5 x - 2\right) \left( 3 x - 4\right)
\dfrac{dy}{dx} = \left(x + 4\right) \left(x + 6\right)
\dfrac{d y}{d x} = \sqrt{x}
\dfrac{d y}{d x} = 18 \sqrt{x}
\dfrac{d y}{d x} = x \sqrt{x}
\dfrac{d y}{d x} = \dfrac{6}{\sqrt{x}}
Find the following indefinite integrals:
\int 4 \ dx
\int x^{3} dx
\int 4 x^{3} dx
\int \dfrac{1}{4} x^{2} dx
\int x^{5} dx
\int \left(x^{2} + 4 x\right) dx
\int \left(5 - 3 t - 4 t^{2}\right) dt
\int \left( 4 x^{3} + 3 x^{2}\right) dx
\int \left( 3 x^{2} + 6 x + 3\right) dx
\int \left( 3 x + 3 x^{2} + x^{3}\right) dx
\int \left(\dfrac{x^{2}}{5} + \dfrac{x^{3}}{4} + 3\right) dx
\int x^{ - 5 } dx
\int \left( 3 x^{ - 2 } - 8 x\right) dx
\int \dfrac{5}{x^{2}} dx
\int x^{\frac{1}{4}} dx
\int x^{\frac{4}{5}} dx
\int 5 x^{\frac{3}{4}} dx
\int \sqrt{x} \ dx
\int \left(x^{\frac{4}{5}} + 5 x^{\frac{2}{3}}\right) dx
\int \left( 5 x^{\frac{7}{3}} + \dfrac{9}{15} x^{\frac{5}{4}}\right) dx
\int \left( 5 x^{\frac{2}{3}} + 3 \sqrt{x} + 6\right) dx
\int \dfrac{1}{\sqrt{x}} \ dx
\int \left( 3 x - 2\right) \left(x + 6\right) dx
\int x \left( 4 x + 7\right) \left( 7 x + 2\right) dx
\int x^{2} \left( 7 x^{4} + 9\right) dx
\int 7 x \left(x - 1\right)^{2} dx
Simplify and hence find the following indefinite integrals:
\int x \left( 7 x^{3} + 3\right)^{2} dx
\int \left(\sqrt{x} - 2\right)^{2} dx
\int \left(\sqrt{x} + \dfrac{5}{x}\right)^{2} dx
\int \sqrt{x} \left( 7 x^{2} + 5 x\right) dx
\int \left(\dfrac{x^{3}}{3} - \dfrac{3}{x^{3}}\right) dx
\int \dfrac{5 x^{5} - 2 x}{x} dx
\int \dfrac{8 x^{6} + 9 x}{x^{5}} dx
\int \dfrac{x^{3} + 5}{\sqrt{x}} dx
\int y \sqrt{y} \ dy
\int \dfrac{1}{y \sqrt{y}} \ dy
\int a y^{3} dy \ , a is a constant
\int \dfrac{1}{n} \sqrt{t} \ dt \ , n is a non-zero constant
For each of the following gradient functions:
State what type of function the antiderivative, f\left(x\right), is.
Find the general antiderivative to f \rq \left( x \right).
Sketch a possible graph for the antiderivative f\left(x\right).
Consider the gradient function f'\left(x\right)=\dfrac{3}{x^{2}}.
Find the general antiderivative to f \rq \left( x \right).
Sketch a possible graph of the antiderivative f\left(x\right).
The gradient function, f'\left(x\right), has only one x-intercept at \left( - 4 , 0\right), a y-intercept at \left(0, - 3 \right) and a constant gradient.
Find f'\left(x\right).
Find the general antiderivative to f \rq \left( x \right).
Sketch a possible graph of the antiderivative f\left(x\right).
Consider the equation \dfrac{d y}{d x} = 4 x + 7.
Find a general equation for y.
Find the equation of y, if the curve passes through the point \left(3, 41\right).
Consider the equation \dfrac{d y}{d x} = 9 x^{2} - 10 x + 2.
Find a general equation for y.
Find the equation of y, if the curve passes through the point \left(2, 13\right).
A family of curves has a gradient function y \rq = 15 x^{2} + 7.
Find the equation of y for the family of curves.
Find the equation of the curve that passes through the point \left(2, 59\right).
Consider the gradient function \dfrac{d y}{d x} = 10 x^{4} + 20 x^{3} + 6 x^{2} + 6 x + 9.
Find an equation for y.
Find the equation of y, if the curve passes through the point \left( - 3 , - 133 \right).
Consider the gradient function \dfrac{d y}{d x} = 9 x^{\frac{2}{3}}.
Find an equation for y.
Find the equation of y, if the curve passes through the point \left(8,\dfrac{889}{5}\right).
Find the equation of a curve p in terms of t, given the following:
- \dfrac{d p}{d t} = 6 t - 5
when t = 3, \dfrac{d p}{d t} = 13 and p = 15
Find the equation of a curve y in terms of x, given the following:
- y \rq \rq = 0.
- y contains the points \left(-1, -1 \right) and \left(0, 2 \right).