Integration: techniques, definite integrals, areas under curves

Pure Mathematics 1 – Integration 📐

1️⃣ Techniques of Integration

Integration is like the opposite of differentiation – you’re “adding up” infinitely many tiny pieces. Below are the most common tricks to solve integrals.

• Substitution (u‑substitution) – Think of changing the variable to make the integral easier, like swapping a messy recipe for a simpler one. If you set u = g(x), then du = g'(x)\,dx and the integral becomes ∫f(g(x))g'(x)\,dx = ∫f(u)\,du.
• Integration by Parts – Uses the product rule in reverse. If ∫u\,dv = uv - ∫v\,du, choose u to simplify after differentiation.
• Partial Fractions – Break a rational function into simpler fractions that are easy to integrate. For example, ∫\frac{1}{x^2-1}\,dx = ∫\Big(\frac{1}{2(x-1)}-\frac{1}{2(x+1)}\Big)\,dx.
• Trigonometric Substitution – Replace x with a trigonometric function to simplify square roots, e.g. x = a\sinθ for √{a^2-x^2}.
• Trigonometric Identities – Use identities like sin^2θ + cos^2θ = 1 to rewrite integrals.

2️⃣ Definite Integrals – The Net Area 📈

A definite integral ∫_{a}^{b} f(x)\,dx gives the net area between the curve f(x) and the x-axis from x = a to x = b. Positive values of f(x) contribute positive area, negative values subtract.

1. Find the antiderivative F(x) of f(x).
2. Apply the Fundamental Theorem of Calculus: ∫_{a}^{b} f(x)\,dx = F(b) - F(a).
3. Interpret the result: if F(b) > F(a), the net area is positive.

Example: ∫_{0}^{π} sin x\,dx = [-cos x]_{0}^{π} = (-cos π) - (-cos 0) = (1) - (-1) = 2. The area under one wave of sine is 2 square units.

3️⃣ Areas Under Curves – From 0 to a Point

To find the area under a curve from x = 0 to x = a, compute ∫_{0}^{a} f(x)\,dx. This is useful for calculating distances, volumes, or probabilities.

📌 Tip: When the curve dips below the x-axis, the integral will subtract that area. If you need the total area regardless of sign, use absolute value: ∫_{a}^{b} |f(x)|\,dx.

4️⃣ Practice Problems 🚀

1. Evaluate ∫_{1}^{4} \frac{1}{x}\,dx.
2. Find the area under f(x) = 3x^2 from x = 0 to x = 2.
3. Use integration by parts to compute ∫ x e^x\,dx.
4. Determine the definite integral ∫_{0}^{π/2} \cos x\,dx and interpret it as an area.

🎉 Keep practicing, and soon you’ll feel as comfortable with integrals as you do with algebraic equations!