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Integral with Steps

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∫ Integral
Integral Type
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Chargement du clavier…

Common Integration

1) Basic Rules

  • The Power Rule:
    ∫xn dx=xn+1n+1+C\int x^n \, dx = \frac{x^{n+1}}{n+1} + C
  • Constant:
    ∫c dx=cx+C\int c \, dx = cx + C
  • Constant Multiple:
    ∫cf(x) dx=c∫f(x) dx\int c f(x) \, dx = c \int f(x) \, dx

2) Special Functions

  • ∫sin⁡(x) dx=−cos⁡(x)+C\int \sin(x) \, dx = -\cos(x) + C
  • ∫cos⁡(x) dx=sin⁡(x)+C\int \cos(x) \, dx = \sin(x) + C
  • ∫ex dx=ex+C\int e^x \, dx = e^x + C
  • ∫1x dx=ln⁡∣x∣+C\int \frac{1}{x} \, dx = \ln|x| + C
Chargement du graphique…
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The Art of Integration: Accumulating Value

Integration is the mathematical process of finding the total of a quantity that is changing. Whether you are calculating the area under a curve, the volume of a complex solid, or the net change of a physical system, our calculator provides the exact symbolic solution through formal algebraic methods.

The Power Rule

∫xn dx=xn+1n+1+C\int x^n \, dx = \frac{x^{n+1}}{n+1} + C

The fundamental rule for all polynomial functions.

u-Substitution

∫f(g(x))g′(x) dx=F(g(x))+C\int f(g(x))g'(x)\,dx = F(g(x)) + C

Transforms composite integrals by substituting u = g(x).

Fundamental Theorem

∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\,dx = F(b) - F(a)

Links antiderivatives to the area under a curve.

How Integration Works

In physics and engineering, integration is used to deduce displacement from velocity, or work done from a varying force. By breaking down a continuous process into infinite infinitesimal slices and summing them, we achieve perfect analytical precision. Our engine handles these complex sums instantly.

Definite vs. Indefinite Integrals

Indefinite Integrals

An indefinite integral represents a family of functions (antiderivatives) whose derivative is the given integrand. Because the derivative of any constant is zero, an indefinite integral must always include a constant of integration, denoted broadly as + C.

∫f(x) dx=F(x)+C\int f(x) \, dx = F(x) + C

Definite Integrals

A definite integral computes a specific, numerical value representing the signed area under a curve between two distinct bounds on the x-axis, typically denoted as lower limit a and upper limit b.

∫abf(x) dx=F(b)−F(a)\int_{a}^{b} f(x) \, dx = F(b) - F(a)

Standard Integration Rules and Formulas

The Power Rule

∫xn dx=xn+1n+1+C\int x^n \, dx = \frac{x^{n+1}}{n+1} + C

The fundamental rule for all polynomial functions.

Exponentials

∫ex dx=ex+C\int e^x \, dx = e^x + C

The base for all growth and decay models.

Frequently Asked Questions

Does this handle second and third derivatives?

Yes! You can calculate higher-order derivatives by simply applying the differentiation operator multiple times.

Are the results accurate?

Our symbolic engine uses formal algebraic rules to guarantee analytical perfection for all supported functions.

How do you solve an integral with steps?

Identify whether the integral is indefinite or definite, then choose the right technique: apply the power rule for polynomials, u-substitution for composite functions, integration by parts for products, or trigonometric identities for trig expressions. For definite integrals, evaluate the antiderivative at both bounds and subtract.

What integration techniques does this calculator support?

Mathsway supports the Power Rule, u-Substitution, Integration by Parts, Trigonometric Integration, Partial Fractions, and standard exponential and logarithmic integral formulas — all with full algebraic steps.

What is the difference between definite and indefinite integrals?

An indefinite integral finds the general antiderivative F(x) + C, representing a family of functions. A definite integral evaluates the accumulated area between a function and the x-axis over a specific interval [a, b], yielding a numerical result via the Fundamental Theorem of Calculus.

How does u-substitution work?

U-substitution is the integral counterpart of the Chain Rule. You substitute u = g(x) to transform a complex integral into a simpler one in terms of u. After integrating, substitute back to express the result in terms of x.

Is this integral calculator free and does it require signup?

Yes, it is completely free, runs directly in your browser with Web Workers, requires no account or signup, and your data remains private on your device.

How are integrals related to derivatives?

By the Fundamental Theorem of Calculus, integration and differentiation are inverse operations. The derivative of an antiderivative gives back the original function. You can verify your work step by step with our derivative calculator with steps.

Can this calculator solve improper integrals?

Improper integrals involve infinite bounds or discontinuous integrands and are evaluated as limits. This calculator focuses on standard definite and indefinite integrals with step-by-step solutions. For limit evaluation, you can also use our limit calculator with steps.