Integration is one of the two fundamental operations of calculus, used to find areas under curves, accumulated quantities, and antiderivatives. Solver AI's integral calculator provides step-by-step solutions for both definite and indefinite integrals, making this challenging topic accessible.
Our calculator handles all major integration techniques: basic antiderivatives, u-substitution, integration by parts, trigonometric integrals, trigonometric substitution, partial fraction decomposition, and improper integrals. It works with polynomial, trigonometric, exponential, logarithmic, and rational functions.
Integration is often considered the most difficult part of calculus because it requires choosing the right technique from many options. Solver AI analyzes each integrand, identifies the optimal method, and explains why that method works for the given function. This builds the intuition needed to tackle new integration problems independently.
Whether you're computing areas and volumes, solving differential equations, or working through physics and engineering applications, Solver AI provides the detailed, step-by-step integration help you need. Type your integral or snap a photo to get started.
The classic integral mistake is forgetting the + C on indefinite integrals. Many students also choose the wrong u in u-substitution, write incorrect limits when changing variables in definite integrals, or apply the power rule when the exponent is -1 (the antiderivative of 1/x is ln|x|, not x⁰/0). Forgetting to multiply by du or by the chain-rule factor when reversing a substitution is another frequent slip that silently turns correct setup into a wrong final answer on exams.
Recognize the form of the integrand first. Polynomial? Use the power rule term by term. Function inside another function with its derivative present? Try u-substitution. Product of unrelated functions? Try integration by parts (LIATE rule for choosing u). Rational function? Factor the denominator and use partial fractions. When you cannot make progress, try rewriting the integrand using trig identities or algebraic manipulation. Practice across many problem types makes the right pattern jump out automatically.
Integrals are the mathematical tool for accumulation. They compute the area under curves, the volume of revolved solids, and the total distance traveled given a velocity function. In probability, integrals turn density functions into actual probabilities. In physics, integrals find work done by varying forces and total charge from current. Engineering, statistics, and economics all use integrals to combine many small contributions into a single total — which is why every quantitative discipline requires fluency with them.
Solution: x⁴/4 + C (power rule for integrals)
Solution: [x² + x]₀² = (4+2) - (0) = 6
Solution: sin²(x)/2 + C (or -cos²(x)/2 + C)
Solution: xeˣ - eˣ + C (integration by parts)
Solver AI supports basic antiderivatives, u-substitution, integration by parts, partial fractions, trigonometric integrals, trigonometric substitution, and improper integrals.
Yes! Solver AI evaluates both definite and indefinite integrals. For definite integrals, it computes the antiderivative and then applies the Fundamental Theorem of Calculus.
Absolutely. Solver AI works with rational functions, trigonometric expressions, exponential functions, logarithmic functions, and compositions thereof.
Yes! Snap a photo of any integral, including those with complex notation like integral signs, limits, and fractions, and Solver AI will provide a step-by-step solution.