Write an equation for f(x), given that the derivative of f \left( x \right) is \dfrac{1}{x}.
Find the primitive function of the following:
Find the following indefinite integrals:
Find the exact value of the following definite integrals:
Determine the integral of the following functions:
f \left( x \right) = \dfrac{2 x^{2} + x - 3}{x}, for x \gt 0.
f \left( x \right) = \dfrac{5 x^{3} + 2 x^{2}}{x^{3}}, for x > 0.
Consider the function f \left( x \right) = x \ln x.
Differentiate f \left( x \right) = x \ln x.
Hence, find \int \ln x \,dx.
Evaluate the following, leaving your answer in the form \dfrac{1}{a} \ln b where a and b are integers between 1 and 10:
\int_{0}^{1} \dfrac{2 x + 1}{3 x^{2} + 3 x + 1} dx
\int_{0}^{2} \dfrac{6 x^{2} - 6 x}{4 x^{3} - 6 x^{2} + 2} dx
Evaluate the following:
\int_{0}^{1} \dfrac{100 p^{4} - 8 p}{5 p^{5} - p^{2} + 2} dp
Leave your answer in the form a \ln b where a and b are integers between 1 and 10.
Evaluate the following, leaving your answer in the form - a \ln b where a and b are integers between 1 and 10:
\int_{0}^{1} \dfrac{- 24 x}{4 x^{2} + 2} dx
\int_{\frac{\pi}{12}}^{\frac{\pi}{4}} \dfrac{4 \cos 2 x}{\sin 2 x} dx
Consider the function f \left( x \right) = \ln \left(x^{2} - 4\right).
Use the chain rule to differentiate f \left( x \right).
Hence, find \int \dfrac{6 x}{x^{2} - 4} \, dx.
Consider the function f \left( x \right) = \ln \left(2x^{2} + 4x\right).
Use the chain rule to differentiate f \left( x \right).
Hence, find \int \dfrac{x+1}{2x^{2} + 4x} \, dx.
Consider the function f \left( x \right) = \ln \left(3x^{3} +5x^2\right).
Use the chain rule to differentiate f \left( x \right).
Hence, find \int \dfrac{18x^{2}+20x}{3x^{3}+5x^{2}} \, dx.
Find the integral of the following functions:
\int \dfrac{4}{4x - 3} \, dx
\int \dfrac{15}{3x - 2} \, dx
\int \dfrac{-8}{2x + 1} \, dx
\int \dfrac{4x}{x^2 + 1} \, dx
\int \dfrac{3 x^2}{x^{3} + 1} \, dx
\int \dfrac{8 x}{2x^{2} - 3} \, dx
\int \dfrac{2 x^2}{2x^{3} + 7} \, dx
\int \dfrac{6 x^2+3}{2x^{3} + 3x} \, dx
Consider the function y = \dfrac{2}{x}.
Sketch a graph of the function for x>0.
Find the exact area bounded by the curve, the x-axis, and the lines x = 2 and x = 3.
Consider the function y = \dfrac{1}{x - 2}.
Sketch a graph of the function for x>0.
Find the exact area bounded by the curve, the x-axis, and the lines x = 3 and x = 5.
Consider the function y = \dfrac{1}{x + 3}.
Sketch a graph of the function y>0.
Find the exact area bounded by the curve, the x-axis, and the lines x = 2 and x = 4.
Consider the function y = \dfrac{1}{2 x + 5}.
Sketch a graph of the function y>0.
Find the exact area enclosed by the curve, the x-axis, and the lines x = 3 and x = 4.
Consider the function y = \dfrac{1}{4 - 3 x}.
Sketch a graph of the function.
Find the exact area enclosed by the curve, the x-axis, and the lines x = 3 and x = 5.
Consider the function y = \dfrac{2}{3 x + 5}.
Sketch a graph of the function x>0.
Find the exact area enclosed by the curve, the coordinate axes and the line x = 4.
Consider the function y = x + \dfrac{1}{x}.
State the domain of the function.
State the value the function approaches as x \to \infty.
Calculate the exact area enclosed by the function, the x-axis, and the lines x = 2 and x = 12.
Find the exact area enclosed by the function f \left( x \right) = x + \dfrac{5}{x + 5}, the x-axis and the lines \\ x = 0 and x = 2.
Consider the function f \left( x \right) = \dfrac{6 x^{5} - 4 x^{2} + x}{2 x^{2}}, for x > 0.
Simplify and hence determine \int f \left( x \right) \, dx.
Find the exact area bounded by f \left( x \right), the x-axis and the lines x = 1 and x = 2.
Consider the functions y = x and y = \dfrac{3}{x}, for x \gt 0.
Sketch the graphs of these functions on the same coordinate axes.
At what x-value do the two functions intersect?
Find the area between the two functions, the x-axis and the line x = 3.
Consider the functions y = - x^{2} and y = \dfrac{1}{4 - x} for 0 \leq x \leq 3.
Sketch the graphs of these functions on the same coordinate axes.
Calculate the exact area bound by the curves between x = 1 and x = 3.
Consider the functions f \left( x \right) = \dfrac{1}{x} and g \left( x \right) = \sqrt{x}.
Sketch the graphs of the two functions on the same coordinate axes.
At what x-value do the two functions intersect?
Find the area of the region bounded by the two curves, the y-axis, and the line y = 2.
Consider the graph of the functions y = x^{2} and y = \dfrac{1}{x}:
Solve for the exact value of k such that: \int_{1}^{k} \dfrac{1}{x} \, dx = \int_{0}^{1} x^2 \, dx
Consider the function f \left( x \right) = \ln \left(x^{2} + 1\right).
Differentiate f \left( x \right).
Hence, find \int \dfrac{x}{x^{2} + 1}\, dx.
Find the area bounded by g \left( x \right) = \dfrac{x}{ \left(x^{2} + 1\right)}, the x-axis and the lines x = 2 and x = 3.
Consider the function f \left( x \right) = \ln \left(\sin \left(x\right) + 1\right).
Differentiate f \left( x \right).
Hence, find \int \dfrac{\cos x}{\sin x + 1} \, dx.
Find the area bounded by g \left( x \right) = \dfrac{\cos x}{\sin \left(x\right) + 1}, the x-axis and the lines x = 0 and x = \dfrac{\pi}{6}.
Find the equation of the curve f \left( x \right), given the derivative function and a point on the curve:
f' \left( x \right) = \dfrac{5}{5 x - 4}, and the point \left(3, \ln 11\right).
f' \left( x \right) = \dfrac{6}{2 x + 5} , and the point \left(1, \ln 49\right).