∫ Integral Calculator: Complete Integration Guide
Master calculus integration with our advanced calculator featuring step-by-step solutions for indefinite and definite integrals.
Integration Techniques Supported
Power Rule
∫x^n dx = x^(n+1)/(n+1) + C
For polynomial integrationU-Substitution
∫f(g(x))g'(x)dx = ∫f(u)du
Reverse chain ruleIntegration by Parts
∫u dv = uv - ∫v du
For products of functionsPartial Fractions
Decompose rational functions
For complex rational expressionsCommon Integrals Reference
| Function | Integral | Function | Integral |
|---|---|---|---|
| Constant (c) | c·x + C | sin(x) | -cos(x) + C |
| x^n (n≠-1) | x^(n+1)/(n+1) + C | cos(x) | sin(x) + C |
| e^x | e^x + C | sec²(x) | tan(x) + C |
| 1/x | ln|x| + C | sec(x)tan(x) | sec(x) + C |
| a^x | a^x/ln(a) + C | csc(x)cot(x) | -csc(x) + C |
Frequently Asked Questions
Q: What is an integral?
A: An integral is the reverse process of differentiation (antiderivative). It represents the area under a curve or accumulation of quantities. Indefinite integrals find general antiderivatives (+C), while definite integrals calculate exact areas between specified bounds. Integrals are fundamental to calculus with applications in physics, engineering, economics, and statistics.
Q: What's the difference between definite and indefinite integrals?
A: Indefinite integrals have no bounds and return a family of functions (antiderivative + C). Example: ∫x²dx = x³/3 + C. Definite integrals have upper and lower limits [a,b] and return a specific numerical value representing the net area. Example: ∫₀¹ x²dx = 1/3. The Fundamental Theorem of Calculus connects both types.
Q: How do you solve integrals?
A: Method depends on function type: Power Rule for polynomials, U-Substitution for composite functions (reverse chain rule), Integration by Parts for products (reverse product rule), Partial Fractions for rational functions, Trigonometric Substitution for radical expressions. Our calculator automatically selects the best method with detailed steps.
Q: What is u-substitution?
A: U-substitution reverses the chain rule. When integrating f(g(x))·g'(x), substitute u=g(x), du=g'(x)dx. Example: ∫2x·cos(x²)dx. Let u=x², du=2xdx. Then ∫cos(u)du = sin(u)+C = sin(x²)+C. This is the most common integration technique for composite functions.
Q: Why add '+C' to indefinite integrals?
A: The constant C represents all possible antiderivatives. Since the derivative of any constant is zero, functions differing only by a constant have identical derivatives. Example: d/dx(x²+5) = 2x, d/dx(x²+100) = 2x. So ∫2xdx = x²+C includes infinitely many solutions. For definite integrals, C cancels out during evaluation.