Is integration by parts in DSE M1?

No. The curriculum notes state that using integration by parts to find indefinite integrals is not required in M1. Substitution is the only technique.

How do I choose the substitution?

Look for an inside function whose derivative also appears as a factor, such as x² + 1 together with x. Let u be the inside function.

Why is the integral of 1/x written with an absolute value?

Because ln x is defined only for positive x, while 1/x exists for negative x too. Writing ln|x| covers both cases.

Indefinite Integration and Integration by Substitution (DSE M1)

Indefinite integration reverses differentiation: ∫ f(x) dx = F(x) + C. M1 needs the formulas for xⁿ, 1/x and eˣ and one technique, integration by substitution. Integration by parts is not in M1.

Solve Indefinite Integration questions with AI

The family P = 20e⁰·³ᵗ + C from Example 1. As C changes the curve moves up and down; only C = 30 starts at P(0) = 50. Try it yourself →

Integration Formulas (DSE M1)

Constant∫ k dx = kx + C

Power∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C (n ≠ −1)

Reciprocal∫ (1/x) dx = ln|x| + C

Exponential∫ eˣ dx = eˣ + C, ∫ e^kx dx = e^kx/k + C

SubstitutionLet u = g(x), du = g′(x) dx

Worked Examples — Indefinite Integration

Example 1: Finding a function from its rate of change

A population P grows at the rate dP/dt = 6e⁰·³ᵗ, and P = 50 when t = 0. Find P in terms of t.

P = ∫ 6e⁰·³ᵗ dt = (6/0.3)e⁰·³ᵗ + C = 20e⁰·³ᵗ + C
When t = 0: 50 = 20 + C, so C = 30

Answer: P = 20e⁰·³ᵗ + 30

Example 2: Integration by substitution

Find ∫ x√(x² + 1) dx.

Let u = x² + 1, so du = 2x dx and x dx = ½ du
∫ ½ u^1/2 du = ½ · (2/3) u^3/2 + C
= (1/3)(x² + 1)^3/2 + C

Answer: (1/3)(x² + 1)^3/2 + C

💡 Exam Traps & Secrets

⚠️ Forgetting + CEvery indefinite integral needs + C. In applied questions the constant is exactly what the initial condition lets you find, so leaving it out loses the rest of the question.

⚠️ Divide by the coefficient of x∫ e⁰·³ᵗ dt = e⁰·³ᵗ / 0.3, not 0.3e⁰·³ᵗ. Integrating divides by the inside coefficient; differentiating multiplies by it.

✏️ Self-Test: Quick Interactive Practice

How to study: Click any question below to instantly load it on the coordinate grid and check your steps.

(Basic formulas)Find the integral of (3x² + 1/x) dx.

(Substitution)Find the integral of 2x e^(x²) dx.

(Finding C)The slope of a curve is dy/dx = 4x − 3 and it passes through (2, 5). Find its equation.