Pure 2 integration
Recognise the integrand, choose a method and check the result.
Start with standard integrals below. Further lessons on substitution, parts, areas, numerical integration and differential equations are being added.
- Standard integralsLearn A-level standard integrals of powers, exponentials, logarithms and trigonometric functions. Explore antiderivative families, check signs and practise exact definite integrals.
- Integrating functions of ax + bIntegrate powers, exponentials and trigonometric functions with linear arguments ax + b. Understand the 1/a factor, logarithmic exceptions, negative slopes and exact definite integrals.
- Integration using trigonometric identitiesIntegrate sin squared, cos squared, tan squared, trigonometric products and fourth powers using identities. Original worked examples, interactive graphs and exact-answer practice.
- Reverse chain rule integrationUse the reverse chain rule to integrate composite powers and exponentials. Identify the inner derivative, match constant factors, check domains and practise definite integrals.
- Logarithmic integralsIntegrate f prime over f using ln absolute value. Learn logarithmic integrals of rational and trigonometric functions, check domain intervals and practise exact definite integrals.
- Integration by substitutionLearn integration by substitution with a manual change-of-variable model. Rewrite every x and dx, handle roots and exponentials, reverse the substitution and check original worked examples.
- Definite integrals by substitutionEvaluate definite integrals by substitution. Change both bounds, track negative differentials and reversed limits, preserve exact answers and check the interval and branch.
- Trigonometric substitution and inverse trig integralsUse x = a sin theta and x = a tan theta to integrate radicals and quadratic denominators. Derive arcsin and arctan integrals, choose valid branches and evaluate exact definite integrals.
- Integration by partsLearn integration by parts from the product rule. Choose u and dv, integrate polynomial-exponential and trig products, handle ln x and evaluate definite integrals with original worked practice.