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Integral Calculator

Compute antiderivatives with step-by-step integration

Select Function Type

Polynomial ax^n Trig sin/cos Exp e^(ax) Reciprocal 1/x

f(x) = a*x^n

intx^dx

Integration Rules

Power rule: int x^n dx = x^(n+1)/(n+1) + C

Exception: int 1/x dx = ln|x| + C

Trig: int cos(x)=sin(x)+C, int sin(x)=-cos(x)+C

Exp: int e^(ax) dx = e^(ax)/a + C

Integration is the reverse of differentiation. The indefinite integral (antiderivative) always includes the constant of integration +C. Definite integrals compute the area under a curve between two bounds.

⚠ Always add +C for indefinite integrals. The constant represents the family of antiderivatives that differ by a vertical shift.

What Is an Integral?

Integration finds the antiderivative and computes area under curves. The power rule integrates x^n by adding 1 to the exponent and dividing. Each function type has specific formulas, and substitution (reverse chain rule) handles composite functions.

Power Rule

int x^n dx = x^(n+1)/(n+1) + C, n != -1. Add 1 to exponent, divide by new exponent.

Trig Integrals

int cos(x)=sin(x)+C, int sin(x)=-cos(x)+C, int sec^2(x)=tan(x)+C. Chain rule: cos(ax)->sin(ax)/a.

Exp Integrals

int e^(ax) dx = e^(ax)/a + C. int a^x dx = a^x/ln(a) + C. The exponential is simple to integrate.

Constant C

The constant of integration C accounts for all possible antiderivatives. For definite integrals, C cancels out.

Teaching Example: int 3x^2 dx = 3 * x^3/3 + C = x^3 + C. Check: d/dx(x^3 + C) = 3x^2. Correct!

Applications

Area Under Curve Volume Physics Probability Engineering

Frequently Asked Questions

How to integrate?▼

Use power rule: int x^n = x^(n+1)/(n+1) + C. Exception: n=-1 gives ln|x| + C. Always add +C.

Power rule for integrals?▼

int x^n dx = x^(n+1)/(n+1) + C for n != -1. Add 1 to exponent, divide by the new exponent.

Why +C?▼

Derivative of any constant is 0, so antiderivatives differ by constants. +C represents the family of all antiderivatives.

Integral of e^x?▼

int e^x dx = e^x + C. int e^(ax) dx = e^(ax)/a + C. The exponential is its own integral up to a factor.

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