Topic 1: Introduction to Integration and Antiderivatives
Easy Questions 🌱
Find the antiderivative of \( f(x) = 2x \).
What is \( \int 5 \, dx \)?
Compute the antiderivative of \( f(x) = x^2 \).
Find \( \int 3x^2 \, dx \).
What is the antiderivative of \( f(x) = 1 \)?
Compute \( \int x \, dx \).
Find the antiderivative of \( f(x) = 4x^3 \).
What is \( \int 2x^3 \, dx \)?
Compute the antiderivative of \( f(x) = x + 1 \).
Find \( \int (x^2 + 2) \, dx \).
Moderate Questions 🌱
Determine the antiderivative of \( f(x) = \frac{1}{x^2} \) (for \( x > 0 \)).
Compute \( \int (3x^2 - 2x + 1) \, dx \).
Find the antiderivative of \( f(x) = \sqrt{x} \).
What is \( \int (x^3 - x) \, dx \)?
Compute the antiderivative of \( f(x) = \frac{2}{x} \) (for \( x > 0 \)).
Find \( \int (4x^2 + 3x - 2) \, dx \).
Determine the antiderivative of \( f(x) = x^{-1/2} \) (for \( x > 0 \)).
Compute \( \int (x^4 - 2x^2 + x) \, dx \).
Find the antiderivative of \( f(x) = 5x^4 - 3x^2 \).
What is \( \int (x^{-2} + x) \, dx \) (for \( x \neq 0 \))?
Hard Questions 🌟
Find the general antiderivative of \( f(x) = \frac{1}{x^3} - \frac{1}{x^2} \) (for \( x > 0 \)).
Compute \( \int (e^x + \sin x) \, dx \) and verify by differentiation.
Determine the antiderivative of \( f(x) = x^2 e^x \) (hint: use parts).
Find \( \int (\cos x - \sin x) \, dx \) and check the result.
Compute the antiderivative of \( f(x) = \frac{1}{1+x^2} \) and relate to a known function.
Topic 2: Indefinite Integrals and Basic Integration Rules
Easy Questions 🌱
Compute \( \int 6x \, dx \).
Find \( \int x^3 \, dx \).
What is \( \int 4 \, dx \)?
Compute \( \int 2x^2 \, dx \).
Find \( \int (x + 3) \, dx \).
What is \( \int 5x^4 \, dx \)?
Compute \( \int (2x - 1) \, dx \).
Find \( \int x^5 \, dx \).
What is \( \int (3x^2 + 1) \, dx \)?
Compute \( \int 7x \, dx \).
Moderate Questions 🌱
Determine \( \int (x^2 - 4x + 2) \, dx \).
Compute \( \int \frac{1}{x^3} \, dx \) (for \( x \neq 0 \)).
Find \( \int (2\sqrt{x} - x) \, dx \).
What is \( \int (x^{-1/3} + 2x) \, dx \) (for \( x > 0 \))?
Compute \( \int (3x^2 - 5x + 4) \, dx \).
Find \( \int \frac{2}{x^2} \, dx \) (for \( x \neq 0 \)).
Determine \( \int (x^{1/2} + x^{-1/2}) \, dx \) (for \( x > 0 \)).
Compute \( \int (x^3 - 2x + 3) \, dx \).
Find \( \int (4x^{-3} + x^2) \, dx \) (for \( x \neq 0 \)).
What is \( \int (5x^4 - 2x^3 + x) \, dx \)?
Hard Questions 🌟
Compute \( \int (e^{2x} + \cos 2x) \, dx \).
Find \( \int (x^2 \ln x) \, dx \) (hint: use integration by parts).
Determine \( \int (\sin 3x - \cos 2x) \, dx \).
Compute \( \int (x e^{-x}) \, dx \) and verify.
Find \( \int (\frac{1}{x} \ln x) \, dx \) (for \( x > 0 \)) using parts.
Topic 3: Integration by Substitution (u-Substitution)
Easy Questions 🌱
Compute \( \int 2x e^{x^2} \, dx \) (let \( u = x^2 \)).
Find \( \int x^2 e^{x^3} \, dx \) (let \( u = x^3 \)).
What is \( \int \cos x \sin^2 x \, dx \) (let \( u = \sin x \))?
Compute \( \int \frac{x}{x^2 + 1} \, dx \) (let \( u = x^2 + 1 \)).
Find \( \int e^{2x} \, dx \) (let \( u = 2x \)).
What is \( \int x e^{x^2 + 1} \, dx \) (let \( u = x^2 + 1 \))?
Compute \( \int \sin x \cos^3 x \, dx \) (let \( u = \cos x \)).
Find \( \int \frac{1}{2x + 1} \, dx \) (let \( u = 2x + 1 \)).
What is \( \int x^3 \sqrt{x^4 + 1} \, dx \) (let \( u = x^4 + 1 \))?
Compute \( \int e^{-x} \sin(e^{-x}) \, dx \) (let \( u = e^{-x} \)).
Moderate Questions 🌱
Determine \( \int \frac{x^2}{\sqrt{x^3 + 1}} \, dx \) (let \( u = x^3 + 1 \)).
Compute \( \int \frac{\sin \sqrt{x}}{\sqrt{x}} \, dx \) (let \( u = \sqrt{x} \)).
Find \( \int x^2 e^{x^3 - 2} \, dx \) (let \( u = x^3 - 2 \)).
What is \( \int \cos^2 x \sin x \, dx \) (let \( u = \cos x \))?
Compute \( \int \frac{x}{(x^2 + 3)^2} \, dx \) (let \( u = x^2 + 3 \)).
Find \( \int x \sin(x^2) \cos(x^2) \, dx \) (let \( u = x^2 \)).
Determine \( \int \frac{e^{\sqrt{x}}}{\sqrt{x}} \, dx \) (let \( u = \sqrt{x} \)).
Compute \( \int \frac{\cos(\ln x)}{x} \, dx \) (let \( u = \ln x \)).
Find \( \int x^2 e^{-x^3} \, dx \) (let \( u = -x^3 \)).
What is \( \int \frac{1}{\sin^2 x \cos x} \, dx \) (let \( u = \sin x \))?
Hard Questions 🌟
Compute \( \int \frac{x^3}{\sqrt{1 - x^4}} \, dx \) (let \( u = 1 - x^4 \)).
Find \( \int \frac{\sin^3 x \cos x}{\sqrt{1 + \sin^4 x}} \, dx \) (let \( u = \sin x \)).
Determine \( \int x^2 e^{x^3} \sin(e^{x^3}) \, dx \) (let \( u = e^{x^3} \)).
Compute \( \int \frac{e^{\tan x} \sec^2 x}{1 + e^{\tan x}} \, dx \) (let \( u = e^{\tan x} \)).
Find \( \int \frac{x^2 + 1}{(x^3 + 3x)^{1/3}} \, dx \) (let \( u = x^3 + 3x \)).
Topic 4: Definite Integrals and the Fundamental Theorem of Calculus
Easy Questions 🌱
Compute \( \int_0^2 3x^2 \, dx \).
Find \( \int_1^2 x \, dx \).
What is \( \int_0^1 4 \, dx \)?
Compute \( \int_0^3 2x \, dx \).
Find \( \int_1^3 x^2 \, dx \).
What is \( \int_0^2 (x + 1) \, dx \)?
Compute \( \int_0^1 5x^3 \, dx \).
Find \( \int_2^4 x^3 \, dx \).
What is \( \int_0^2 (2x^2 + 1) \, dx \)?
Compute \( \int_1^2 3x^4 \, dx \).
Moderate Questions 🌱
Determine \( \int_0^1 (x^2 - 2x) \, dx \).
Compute \( \int_1^2 \frac{1}{x^2} \, dx \).
Find \( \int_0^1 \sqrt{x} \, dx \).
What is \( \int_0^2 (x^3 - x) \, dx \)?
Compute \( \int_1^3 (2x - 3) \, dx \).
Find \( \int_0^1 x^{-1/2} \, dx \).
Determine \( \int_0^2 (x^2 + 2x + 1) \, dx \).
Compute \( \int_1^4 \frac{2}{x} \, dx \).
Find \( \int_0^1 (x^3 - 2x^2) \, dx \).
What is \( \int_2^3 (x^{-2} + x) \, dx \)?
Hard Questions 🌟
Compute \( \int_0^1 e^x \, dx \) and verify using the Fundamental Theorem.
Find \( \int_0^{\pi} \sin x \, dx \).
Determine \( \int_1^2 x \ln x \, dx \) using integration by parts.
Compute \( \int_0^{\pi/2} \cos^2 x \, dx \) using a trigonometric identity.
Find \( \int_0^1 \frac{1}{1+x^2} \, dx \) and relate to arctangent.
Topic 5: Area Under a Curve and Between Two Curves
Easy Questions 🌱
Find the area under \( y = x \) from \( x = 0 \) to \( x = 2 \).
Compute the area under \( y = x^2 \) from \( x = 0 \) to \( x = 1 \).
What is the area under \( y = 3 \) from \( x = 1 \) to \( x = 4 \)?
Find the area under \( y = 2x \) from \( x = 0 \) to \( x = 3 \).
Compute the area under \( y = x^3 \) from \( x = 0 \) to \( x = 2 \).
What is the area under \( y = 4 - x \) from \( x = 0 \) to \( x = 2 \)?
Find the area under \( y = x^2 + 1 \) from \( x = 0 \) to \( x = 1 \).
Compute the area under \( y = \sqrt{x} \) from \( x = 0 \) to \( x = 4 \).
What is the area under \( y = 2x^2 \) from \( x = 0 \) to \( x = 2 \)?
Find the area under \( y = 1/x \) from \( x = 1 \) to \( x = 2 \) (for \( x > 0 \)).
Moderate Questions 🌱
Determine the area between \( y = x \) and \( y = x^2 \) from \( x = 0 \) to \( x = 1 \).
Compute the area between \( y = x^2 \) and \( y = 2x \) from \( x = 0 \) to \( x = 2 \).
Find the area between \( y = x^3 \) and \( y = x \) from \( x = -1 \) to \( x = 1 \).
What is the area between \( y = \sin x \) and \( y = 0 \) from \( x = 0 \) to \( x = \pi \)?
Compute the area between \( y = x^2 - 1 \) and \( y = 0 \) from \( x = -1 \) to \( x = 2 \).
Find the area between \( y = 2x - x^2 \) and \( y = x \) from \( x = 0 \) to \( x = 1 \).
Determine the area between \( y = \sqrt{x} \) and \( y = x^2 \) from \( x = 0 \) to \( x = 1 \).
Compute the area between \( y = \cos x \) and \( y = \sin x \) from \( x = 0 \) to \( x = \pi/2 \).
Find the area between \( y = x^3 - x \) and \( y = 0 \) from \( x = -1 \) to \( x = 1 \).
What is the area between \( y = 1/x \) and \( y = x \) from \( x = 1 \) to \( x = 2 \)?
Hard Questions 🌟
Compute the area between \( y = e^x \) and \( y = e^{-x} \) from \( x = 0 \) to \( x = 1 \).
Find the area between \( y = \sin^2 x \) and \( y = \cos^2 x \) from \( x = 0 \) to \( x = \pi \).
Determine the area between \( y = x^2 \ln x \) and \( y = 0 \) from \( x = 1 \) to \( x = e \).
Compute the area between \( y = \frac{1}{1+x^2} \) and \( y = x \) from \( x = 0 \) to \( x = 1 \).
Find the area between \( y = \sqrt{1 - x^2} \) and \( y = x^2 - 1 \) from \( x = -1 \) to \( x = 1 \).
Topic 6: Applications of Integration (Accumulated Change, Average Value)
Easy Questions 🌱
Find the accumulated change of \( v(t) = 2t \) from \( t = 0 \) to \( t = 3 \).
Compute the total distance for \( v(t) = 3t \) from \( t = 0 \) to \( t = 2 \).
What is the average value of \( f(x) = 4 \) on \( [0, 2] \)?
Find the accumulated change of \( v(t) = t^2 \) from \( t = 0 \) to \( t = 1 \).
Compute the average value of \( f(x) = x \) on \( [0, 2] \).
What is the total distance for \( v(t) = 4t \) from \( t = 0 \) to \( t = 1 \)?
Find the accumulated change of \( v(t) = 3 \) from \( t = 1 \) to \( t = 4 \).
Compute the average value of \( f(x) = x^2 \) on \( [0, 1] \).
What is the total distance for \( v(t) = 2t^2 \) from \( t = 0 \) to \( t = 2 \)?
Find the average value of \( f(x) = 2x \) on \( [0, 3] \).
Moderate Questions 🌱
Determine the accumulated change of \( v(t) = t^3 - t \) from \( t = 0 \) to \( t = 2 \).
Compute the average value of \( f(x) = \sqrt{x} \) on \( [0, 4] \).
Find the total distance for \( v(t) = \sin t \) from \( t = 0 \) to \( t = \pi \).
What is the accumulated change of \( v(t) = e^t \) from \( t = 0 \) to \( t = 1 \)?
Compute the average value of \( f(x) = x^3 \) on \( [0, 2] \).
Find the accumulated change of \( v(t) = \cos t \) from \( t = 0 \) to \( t = 2\pi \).
Determine the total distance for \( v(t) = 1/t \) from \( t = 1 \) to \( t = 2 \) (for \( t > 0 \)).
Compute the average value of \( f(x) = \sin x \) on \( [0, \pi] \).
Find the accumulated change of \( v(t) = t^2 + 1 \) from \( t = -1 \) to \( t = 1 \).
What is the average value of \( f(x) = e^x \) on \( [0, 1] \)?
Hard Questions 🌟
Compute the total distance traveled if \( v(t) = t \sin t \) from \( t = 0 \) to \( t = \pi \).
Find the average value of \( f(x) = x \ln x \) on \( [1, e] \).
Determine the accumulated change of \( v(t) = e^{-t} \cos t \) from \( t = 0 \) to \( t = \pi \).
Compute the average value of \( f(x) = \frac{1}{1+x^2} \) on \( [0, 1] \).
Find the total distance for \( v(t) = \sqrt{1 + t^2} \) from \( t = 0 \) to \( t = 1 \).
Topic 7: Volumes of Solids (Disk, Washer, and Shell Methods)
Easy Questions 🌱
Find the volume of the solid obtained by rotating \( y = x \) around the x-axis from \( x = 0 \) to \( x = 1 \) (disk method).
Compute the volume of \( y = x^2 \) rotated around the x-axis from \( x = 0 \) to \( x = 1 \).
What is the volume of \( y = 2 \) rotated around the x-axis from \( x = 0 \) to \( x = 1 \)?
Find the volume of \( y = x^3 \) rotated around the x-axis from \( x = 0 \) to \( x = 2 \).
Compute the volume of \( y = \sqrt{x} \) rotated around the x-axis from \( x = 0 \) to \( x = 4 \).
What is the volume of \( y = x + 1 \) rotated around the x-axis from \( x = 0 \) to \( x = 1 \)?
Find the volume of \( y = x^2 + 1 \) rotated around the x-axis from \( x = 0 \) to \( x = 1 \).
Compute the volume of \( y = 2x \) rotated around the x-axis from \( x = 0 \) to \( x = 2 \).
What is the volume of \( y = 1/x \) rotated around the x-axis from \( x = 1 \) to \( x = 2 \)?
Find the volume of \( y = x^2 \) rotated around the y-axis from \( x = 0 \) to \( x = 1 \) (shell method).
Moderate Questions 🌱
Determine the volume of the solid formed by rotating \( y = x^2 \) and \( y = x \) around the x-axis from \( x = 0 \) to \( x = 1 \) (washer method).
Compute the volume of \( y = x \) and \( y = x^2 \) rotated around the y-axis from \( y = 0 \) to \( y = 1 \).
Find the volume of \( y = \sin x \) rotated around the x-axis from \( x = 0 \) to \( x = \pi \).
What is the volume of \( y = x^3 \) and \( y = x \) rotated around the x-axis from \( x = 0 \) to \( x = 1 \)?
Compute the volume of \( y = 2 - x \) and \( y = x \) rotated around the x-axis from \( x = 0 \) to \( x = 1 \).
Find the volume of \( y = x^2 \) and \( y = 2x \) rotated around the y-axis from \( x = 0 \) to \( x = 2 \).
Determine the volume of \( y = \sqrt{x} \) and \( y = x^2 \) rotated around the x-axis from \( x = 0 \) to \( x = 1 \).
Compute the volume of \( y = e^x \) rotated around the x-axis from \( x = 0 \) to \( x = 1 \).
Find the volume of \( y = x^2 - 1 \) and \( y = 0 \) rotated around the x-axis from \( x = -1 \) to \( x = 1 \).
What is the volume of \( y = \cos x \) rotated around the x-axis from \( x = 0 \) to \( x = \pi/2 \)?
Hard Questions 🌟
Compute the volume of the solid formed by rotating \( y = e^x \) and \( y = e^{-x} \) around the x-axis from \( x = 0 \) to \( x = 1 \).
Find the volume of \( y = \sin x \) and \( y = \cos x \) rotated around the x-axis from \( x = 0 \) to \( x = \pi/2 \).
Determine the volume of \( y = x^2 \ln x \) rotated around the x-axis from \( x = 1 \) to \( x = e \).
Compute the volume of \( y = \frac{1}{1+x^2} \) and \( y = x \) rotated around the x-axis from \( x = 0 \) to \( x = 1 \).
Find the volume of \( y = \sqrt{1 - x^2} \) rotated around the y-axis from \( x = -1 \) to \( x = 1 \) (use shell method).
Topic 8: Improper Integrals
Easy Questions 🌱
Compute \( \int_1^{\infty} \frac{1}{x^2} \, dx \).
Find \( \int_0^{\infty} e^{-x} \, dx \).
What is \( \int_2^{\infty} \frac{1}{x^3} \, dx \)?
Compute \( \int_0^{\infty} e^{-2x} \, dx \).
Find \( \int_1^{\infty} \frac{1}{x^4} \, dx \).
What is \( \int_0^{\infty} e^{-3x} \, dx \)?
Compute \( \int_3^{\infty} \frac{1}{x^2} \, dx \).
Find \( \int_0^{\infty} e^{-x/2} \, dx \).
What is \( \int_1^{\infty} \frac{1}{x^{3/2}} \, dx \)?
Compute \( \int_0^{\infty} e^{-4x} \, dx \).
Moderate Questions 🌱
Determine \( \int_0^{\infty} \frac{1}{1 + x^2} \, dx \).
Compute \( \int_1^{\infty} \frac{\ln x}{x^2} \, dx \).
Find \( \int_0^{\infty} x e^{-x^2} \, dx \).
What is \( \int_0^{\infty} \frac{\sin x}{x} \, dx \)?
Compute \( \int_1^{\infty} \frac{1}{x \ln x} \, dx \).
Find \( \int_0^{\infty} e^{-x} \cos x \, dx \).
Determine \( \int_0^{\infty} \frac{x}{1 + x^4} \, dx \).
Compute \( \int_1^{\infty} \frac{1}{\sqrt{x} (1 + x)} \, dx \).
Find \( \int_0^{\infty} x^2 e^{-x} \, dx \).
What is \( \int_0^{\infty} e^{-x} \sin 2x \, dx \)?
Hard Questions 🌟
Compute \( \int_0^{\infty} \frac{x^2}{(x^2 + 1)^2} \, dx \).
Find \( \int_0^{\infty} \frac{\ln x}{1 + x^2} \, dx \).
Determine \( \int_{-\infty}^{\infty} \frac{1}{x^2 + 2x + 2} \, dx \).
Compute \( \int_0^{\infty} e^{-x^2} \cos(2x) \, dx \).
Find \( \int_0^{\infty} \frac{\sin^2 x}{x^2} \, dx \).