Mathematics

A-Level Integration: From Antiderivatives to Areas

An intuitive exploration of integration covering power rules, substitution, integration by parts, definite integrals as areas, and standard function results.

\int x^n dx = \frac{x^{n+1}}{n+1} + C

Think of integration as unwinding the process of differentiation. If differentiation tells us the slope, integration asks us to recover the original function from that slope.

Step-by-step solver

(a) Basic Integration

Use the reverse power rule on each term of the polynomial.

\int (3x^2+4x-5) dx = x^3 + 2x^2 - 5x + C

(b) Substitution

Identify the inner function u and substitute, noting that du accounts for the derivative factor.

\int 2x(x^2+1)^5 dx = \frac{(x^2+1)^6}{6} + C

Assign u and dv according to LIATE, then apply the formula.

\int x e^x dx = x e^x - e^x + C

(d) Definite Integral

Evaluate the antiderivative at the upper limit 3 and subtract the evaluation at 1.

[\frac{x^3}{3}]_1^3 = \frac{27}{3} - \frac{1}{3} = \frac{26}{3}

(e) Standard Functions

Apply standard integral results for each function type individually.

\int (\sin x + e^{2x} + \frac{1}{x}) dx = -\cos x + \frac{1}{2}e^{2x} + \ln|x| + C

Original question

A-Level integration explained

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