Find the primitive of the following:
Find the following indefinite integrals:
Find the exact value of the following definite integrals:
Consider the function y = x \ln x.
Differentiate the function.
Evaluate \int_{2}^{3} \ln x \, dx, correct to two decimal places.
Consider the function y=x \sin x.
Find \dfrac{d}{dx} \left( x \sin x\right).
Hence find \int 4 x \cos x \,dx.
Consider the function y = e^{x} + e^{ - x }.
Differentiate the function.
Hence find the exact value of \int_{0}^{3} \dfrac{e^{x} - e^{ - x }}{e^{x} + e^{ - x }} dx.
Consider the function y = e^{ 3 x} \left(x - \dfrac{1}{3}\right).
Find y'.
Hence find the exact value of \int_{6}^{9} x e^{ 3 x} dx.
Calculate the exact area bounded by:
The curve y = e^{\frac{x}{4}} + e^{ - \frac{x}{4} }, the coordinate axes, and the line x = \dfrac{1}{2}.
The curve y = 3 \left(\sin x + 1\right), the x-axis and the lines x = 0 and x = 2 \pi.
The curve f \left( x \right) = x + \dfrac{5}{x + 5}, the x-axis and the lines x = 0 and x = 2.
The curve y = \dfrac{\cos x}{1 + \sin x}, the x-axis and the lines x = 0 and x = \dfrac{\pi}{6}.
The curve y = 2 + \dfrac{1}{4} \sin 2 x, the x-axis and the lines x = 0 and x = \dfrac{\pi}{2}.
The curve y = \dfrac{1}{3} + 3 \cos x, the x-axis and the lines x = - \dfrac{\pi}{6} to x = \dfrac{\pi}{6}.
The curve y = e^{1 - x}, the line y = x and the line x = 3.
The curves y = e^{x} and y = e^{ - x }, and the line x = 3.
The curves y = e^{ 5 x} and y = e^{ - 5 x }, and the lines x = - 2 and x = 2.
The curves y = \sin 2 x and y = \cos x over the domain \dfrac{\pi}{6} \leq x \leq \dfrac{\pi}{2}.
The curves y = e^{ 2 x} and y = e^{ - x }, the x-axis, and the lines x = - 2 and x = 2.
Consider the integral \int_{0}^{\frac{\pi}{2}} \left(2 - \cos x\right) dx.
Sketch the graph of y = 2 - \cos x, clearly showing the area represented by \int_{0}^{\frac{\pi}{2}} \left(2 - \cos x\right) dx.
Hence find the exact area represented by the integral \int_{0}^{\frac{\pi}{2}} \left(2 - \cos x\right) dx.
Consider the functions y = \left|x\right| and y = \dfrac{2}{x}.
Sketch the graphs of these functions on the same number plane.
Find the exact x-value of the point of intersection of the two functions.
Calculate the exact area between the two curves, the x-axis and the line x = 5.
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 number plane.
Calculate the exact area bound by the curves between x = 1 and x = 3.
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.
Consider the functions f \left( x \right) = e^{x} + 12 and g \left( x \right) = 4 x + e^{3}.
Find the x-coordinate of the point of intersection that is to the right of the origin.
Calculate the exact area in the first quadrant bounded by the two curves and the y-axis.
Consider the functions y = e^{\frac{x}{6}} and y = e^{\frac{3}{2}}.
Find the x-coordinate of the point of intersection.
Calculate the exact area bounded by y = e^{\frac{x}{6}}, the y-axis and the line y = e^{\frac{3}{2}}.
Consider the functions f \left( x \right) = 3 - e^{x} and g \left( x \right) = e^{x} + 1 .
Find the x-coordinate of the point of intersection.
Calculate the exact area bound between the two curves and the line x = \ln 3.
Given that \int_{3}^{5} \dfrac{e^{x}}{1 + e^{x}} dx = \ln c, find the value of c.
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
Prove that \left(1 + \tan ^{2}x\right) \cos ^{2}x = 1.
Hence, find the exact value of \int_{\frac{\pi}{8}}^{\frac{\pi}{5}} \left(1 + \tan ^{2}x\right) \cos ^{2}x dx.
Find the function h \left( x \right), given that h(0) = \ln 8 and h' \left( x \right)=\dfrac{\cos x \sin x}{\sin ^{2}\left(x\right) + 8}.
Consider the function f \left( x \right) = e^{ 3 x} - e^{ - 3 x }.
Find the x-intercept of the function.
Find the limiting function as x approaches +\infty.
Show that the function is odd.
Find the exact area bounded by the curve, the coordinate axes, and the line x = 3.
Find the exact area bound by the curve, the x-axis, and the lines x = 1 and x = - 1.
The cross-section of a satellite dish can be estimated by the area bound by the hyperbola y = \dfrac{32}{x} \,and the line y = - x + 12 \,, as shown in the diagram:
Find the x-values of the points of intersection of the hyperbola and the line.
Hence determine the cross-sectional area of the satellite dish, correct to two decimal places.