Basic integrals
Start with powers, sums and constant factors. Differentiate to check your antiderivative.
Practice integrals: exercises with hints and solutions
Try solving on your own and use only the help you need. In Learn, choose a method, adjust difficulty, check your answer and load the exercise in the calculator to study the solution.
How to organize your practice
- Choose a family. Start with basic rules if you are still building confidence with derivatives and immediate antiderivatives.
- Choose a fixed level or level progression, which increases every six exercises. Adaptive difficulty changes based on your answers.
- Enter an antiderivative for indefinite integrals and an exact value for definite or multiple integrals. Use pi, fractions and functions such as sin(x).
- Check, review the mistake and try again. Hints and solutions are available; exam mode hides feedback until you finish.
Progress is saved on this device without an account. In polar, cylindrical and spherical practice problems, the Jacobian is already included: do not multiply by it twice.
Integral exercises by method and coordinates
Integration by substitution
Recognize a composite function and its derivative. Transform the differential and bounds too.
Integration by parts
Choose u and dv in products involving logarithms, polynomials, exponentials or trigonometric functions.
Rational integrals and partial fractions
Compare degrees, divide when needed and distinguish linear, repeated and quadratic factors.
Trigonometric integrals
Work with powers of sine and cosine and choose the identity that simplifies the integral.
Integrals with radicals
Use trigonometric substitutions while tracking intervals and signs of radicals.
Double integrals
Describe a plane region, interpret variable bounds and integrate from inside out.
Triple integrals
Set up volumes and spatial regions. University-level exercises include radical bounds.
Integrals in polar coordinates
Practice disks, sectors and curved regions, including the r factor in the area element.
Integrals in cylindrical coordinates
Progress from cylinders to solids between paraboloids and intersections of cones and spheres.
Integrals in spherical coordinates
Work with spheres and cones. Distinguish the angles and the r² sin φ Jacobian.
Six starter exercises with explained answers
Try them before opening the answer. If you get stuck, first identify the variable, bounds and pattern in the integrand.
1. Powers and linearity
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Integrate each term separately. Differentiating the answer recovers all three original terms.
Study this method step by step →2. Substitution
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Take u = x² and du = 2x dx. Integrate cos u, then return to x.
Study this method step by step →3. Integration by parts
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Choose u = x and dv = eˣ dx. The formula gives xeˣ − ∫eˣ dx.
Study this method step by step →4. Definite integral
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An antiderivative is x³/3. Evaluate it at 2 and subtract its value at 0.
Study this method step by step →5. Triangular region
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The first stage gives (1 − y²)/2. Then integrate from y = 0 to y = 1.
Study this method step by step →6. Volume in cylindrical coordinates
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The r factor is already the Jacobian. Integrating in z gives 3r; the next stages give 6 and then 12π.
Study this method step by step →University integral practice
The multivariable paths include a University level. Work with dependent bounds, radicals, solids between paraboloids and intersections of cones and spheres. Before calculating, identify which surface bounds each variable and check radical domains.
Use Visualize in 3D to explore the region, then return to the exercise. A figure helps interpret the setup but does not replace checking bounds and integration conditions.
Review cylindrical coordinates →Continue with IntegralPaso
Use the calculator to study a specific solution, the library to understand the method and Learn to practice before revealing the answer.