Substitution Methods
Evaluate the following integral.
\int \frac{\cos x}{\sin x} dx
What is \ln |sin x| + C?
Evaluate the following integral.
\int \sin^2 \cos^3 dx
What is \frac 1 3 \sin^3 x - \frac 1 5 \sin^5 x + C?
Evaluate the following integral.
\int_1^\infty \frac 1 {x^2} dx
What is 1?
Compute the area bounded by y = 2x + 3, y = 0, x = 0, and x = 2.
What is 10 square units?
Compute the mass of the one-dimensional rod with linear density \rho(x) = e^x on [0, \ln 5].
What is 4 units?
Evaluate the following integral.
\int_0^{\ln 2} e^x (e^x + 1)^e dx
What is \frac{3^{e + 1}}{e + 1} - \frac{2^{e + 1}}{e + 1}?
Evaluate the following integral.
\int \frac{\tan^4 x}{(1 - \cos^2 x)^2} dx
What is \frac 1 3 \tan^3 x + \tan x + C?
Evaluate the following integral.
\int_2^\infty \frac 1 {x^2 - 1} dx
What is \frac 1 2 \ln 3?
Compute the area bounded by y = (1 + x^2)^{-1}, y = 0, x = 0, and x = 1.
What is \pi/4 square units?
Compute the work required to raise a 1-kg bucket tied to a weightless rope a vertical distance of 10 m if the bucket contains an initial volume of 5 kg of water and water leaks from the bucket at a constant rate of 0.05 kg / m.
What is 563.5 joules?
Evaluate the following integral.
\int x \ln^2 x dx
What is \frac 1 2 x^2 \ln^2 |x| - \frac 1 2 \ln |x| + \frac 1 4 x^2 + C?
Evaluate the following integral.
\int_0^{1/\sqrt 3} \sqrt{x^2 + 1} dx
What is \frac 1 3 + \frac 1 4 \ln 3?
Evaluate the following integral.
\int_0^\infty \frac{x^3}{x^2 + 1} dx
What is diverges?
Compute the volume of the solid of revolution obtained by rotating the region bounded by y = x^3 and y = x^2 about the x-axis.
What is \frac{2 pi}{35} cube units?
Compute the work required to pump water from a cylindrical tank of height 8 m and radius 2 m through an outlet pipe 2 m above the top of the tank if the tank is half-full.
What is 1254400 \pi joules?
Evaluate the following integral.
\int_0^1 x^3 \sqrt{1 - x^2} dx
What is 2/15?
Evaluate the following integral.
\int \frac{\sqrt{1 - x^2}}{x^2} dx
What is -\frac{\sqrt{1 - x^2}} x - \sin^{-1} x + C?
Compute the volume of the solid of revolution obtained by rotating the region bounded by y = x^{-1} and y = 0 defined for x \ge 1 about the x-axis.
What is pi?
Compute the volume of the solid of revolution obtained by rotating the region bounded by y = x^3 and y = x^2 about the y-axis.
What is \frac{\pi}{10} cube units?
Compute the work required to pump water from a tank shaped like an inverted right-circular cone of height 6 m and radius 1.5 m through an outlet pipe 4 m above the top of the tank if the tank is full.
What is 242550 \pi joules?
Evaluate the following integral.
\int \frac{e^{4x}}{(1 - e^{2x})^2} dx
What is \frac 1 2 [\ln(e^{2x} - 1) - \frac 1 {e^{2x} - 1}] + C?
Evaluate the following integral.
\int \frac 1 {x^2 (x^2 - 4) dx
What is \frac 1 8 \ln |\frac{x - 2}{\sqrt{x^2 - 4}}| + \frac 1 {4x} + C?
Compute the volume of the solid of revolution obtained by rotating the region bounded by y = x^{-1} and y = 0 defined for x \ge 1 about the y-axis.
What is infinite?
Compute the volume of the solid of revolution obtained by rotating the region bounded by y = \sin x and y = x about the x-axis on the interval [-pi, pi].
What is \frac 2 3 \pi^4 - \pi^2 cube units?
Compute the work required to fill a spherical tank of radius 8 m with water through a 2 m outflow pipe located at the top of the sphere.
What is 66901.3 pi joules?