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A Levels / Maths

Integration

Learn A-Level integration, including integration rules, definite integrals, area under curves, integration by substitution, integration by parts, partial fractions and separable differential equations.

Integration is the inverse process of differentiation and is used to find antiderivatives, accumulated change and areas under curves.
The Fundamental Theorem of Calculus connects differentiation and integration and allows definite integrals to be evaluated using an antiderivative.
The integration power rule is ∫xn dx=xn+1n+1+C\int x^n\,dx=\frac{x^{n+1}}{n+1}+C for n≠−1n\neq-1, where CC is the constant of integration.
The integral of 1/x1/x is ∫1x dx=ln⁡∣x∣+C\int\frac{1}{x}\,dx=\ln|x|+C, which is an important exception to the usual power rule.
Exponential functions can be integrated using ∫ekx dx=1kekx+C\int e^{kx}\,dx=\frac{1}{k}e^{kx}+C for non-zero constant kk.
Trigonometric integration includes ∫sin⁡(kx) dx=−1kcos⁡(kx)+C\int\sin(kx)\,dx=-\frac{1}{k}\cos(kx)+C and ∫cos⁡(kx) dx=1ksin⁡(kx)+C\int\cos(kx)\,dx=\frac{1}{k}\sin(kx)+C.
A definite integral ∫abf(x) dx\int_a^b f(x)\,dx is evaluated using F(b)−F(a)F(b)-F(a), where F′(x)=f(x)F'(x)=f(x).
Definite integrals can be used to calculate the area under a curve, with care taken when a curve lies below the x-axis.
The area between two curves is found by integrating the difference between the upper and lower functions over the required interval.
Integration can be understood as the limit of a sum, where increasingly narrow strips approximate the total area beneath a curve.
Integration by substitution simplifies an integral by replacing a suitable expression with a new variable and is the reverse process of the chain rule.
Substitution rule integrals require choosing a substitution that transforms the original expression into a form that can be integrated directly.
Integration by parts is the reverse process of the product rule and uses the formula ∫u dv=uv−∫v du\int u\,dv=uv-\int v\,du.
Partial fractions can be used before integration to split suitable rational expressions with linear denominator factors into simpler fractions.
Separable first-order differential equations are solved by separating the variables, integrating both sides and then using any given condition to find a particular solution and interpret it in context.
A Level Integration | Substitution, By Parts & Integrals