Integral Calculator

Integration is the inverse operation of differentiation. An antiderivative F(x) of f(x) satisfies F'(x) = f(x). Every continuous function has infinitely many antiderivatives, differing by a constant. **Reverse Power Rule**: ∫ ax^n dx = (a/(n+1)) × x^(n+1) + C, provided n ≠ -1 In plain language: increase the exponent by 1, then divide by the new exponent. **The Exception (n = -1)**: When n = -1, this formula breaks (division by zero). Instead: ∫ (1/x) dx = ln|x| + C. **Examples**: - ∫ x dx = x²/2 + C - ∫ x² dx = x³/3 + C - ∫ 3x² dx = x³ + C - ∫ 6x⁵ dx = x⁶ + C - ∫ x^(1/2) dx = (2/3)x^(3/2) + C - ∫ dx = x + C (integral of 1) - ∫ 0 dx = C (integral of 0) **The +C (Constant of Integration)**: Because the derivative of any constant is 0, when we 'undo' a derivative, we lose information about any constant that was there. So every antiderivative has an arbitrary +C. When we have additional info (an initial condition), we can solve for C. **Definite Integrals**: ∫[a,b] f(x) dx = F(b) - F(a). This computes the net area under the curve between x = a and x = b. The constant C cancels out, so no +C appears in definite integrals. **The Fundamental Theorem of Calculus**: This connects differentiation and integration. Part 1 says that if F(x) = ∫[a,x] f(t) dt, then F'(x) = f(x). Part 2 says that if F'(x) = f(x), then ∫[a,b] f(x) dx = F(b) - F(a). Together, they make integration practical. **Geometric Interpretation**: For positive functions, the definite integral equals the area between the curve and the x-axis. For functions that go negative, it gives net area (positive areas minus negative areas). This has profound implications — if velocity is v(t), then ∫[0,T] v(t) dt is the total displacement from t=0 to t=T. **Applications**: 1. **Physics**: If a(t) is acceleration, then ∫a(t)dt gives velocity, and ∫v(t)dt gives position. Free fall from rest: v(t) = ∫9.8 dt = 9.8t + C. Since v(0) = 0, C = 0, so v(t) = 9.8t. Then s(t) = ∫9.8t dt = 4.9t² + C. If s(0) = 0, then s(t) = 4.9t². 2. **Area Between Curves**: ∫[a,b] (f(x) - g(x)) dx gives the area between two curves from x = a to x = b. 3. **Volume of Revolution**: Rotating a curve around an axis and computing the resulting solid's volume uses integration. 4. **Work**: Work = ∫F dx, where F is force and dx is distance. If force is variable (like a spring), integration is necessary. 5. **Average Value**: The average value of f(x) on [a,b] is (1/(b-a)) × ∫[a,b] f(x) dx. **Integration Techniques for More Complex Functions**: - **Substitution (u-substitution)**: Reverses the chain rule - **Integration by parts**: Reverses the product rule - **Partial fractions**: For rational functions - **Trigonometric substitution**: For expressions with √(a²-x²), √(a²+x²) - **Numerical methods**: Simpson's rule, trapezoidal rule when analytical solutions don't exist