Compute antiderivatives with step-by-step integration
Select Function Type
f(x) = a*x^n
intx^dx
f(x) = sin(ax) or cos(ax)
(x) dx
f(x) = e^(ax)
int e^(x) dx
f(x) = 1/(ax)
int 1/(x) dx
Result
Integral
Derivation
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 CurveVolumePhysicsProbabilityEngineering
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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