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Calculate indefinite integrals (antiderivatives) of polynomial functions with step-by-step solutions. Apply the power rule for integration with detailed working.
An integral (antiderivative) is the reverse of differentiation, finding a function whose derivative equals the given function. It represents accumulated area under a curve.
Use x as the variable, ^ for exponents
| f(x) | ∫f(x)dx |
|---|---|
| x^n | x^(n+1)/(n+1) + C |
| 1/x | ln|x| + C |
| e^x | e^x + C |
| sin(x) | -cos(x) + C |
| cos(x) | sin(x) + C |
| sec²(x) | tan(x) + C |
| 1/(1+x²) | arctan(x) + C |
Formula
∫x^n dx = x^(n+1) / (n+1) + C (n ≠ −1)x = The variable of integration
n = The exponent of x
C = Constant of integration
Worked Example
Integrate f(x) = 6x² + 4x − 3
Did you know? The integral sign ∫ was introduced by Leibniz in 1675 and is an elongated 'S' from the Latin word 'summa' (sum), reflecting that integration is a continuous summation process (source: Mathematical Association of America).
Sources
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