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Integration Techniques

Learn integration properly: substitution, integration by parts, partial fractions, and definite integrals with physics-friendly worked examples.

  • University Physics Year 1
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Integration is the inverse operation of differentiation, and it is also how you compute “accumulated total” quantities:

  • displacement from velocity: x(t)-x(0) = ∫₀^t v(t') dt'
  • work from force: W = ∫ vector F · d vector ℓ
  • charge from current: q(t)-q(0) = ∫₀^t i(t') dt'
Why physics needs this (examples you can click)

1) What an integral means

An indefinite integral is an antiderivative:

∫ f(x) dx = F(x) + C where F'(x) = f(x).

A definite integral is a number:

∫ₐ^b f(x) dx.

Fundamental theorem of calculus:

∫ₐ^b f(x) dx = F(b)-F(a) if F'(x) = f(x).
The #1 habit that makes you good at integration

Always differentiate your final answer once. If you don’t get the integrand back, something is wrong.

2) Core techniques

Integration workflow (what to try in order)
  1. Simplify first: factor constants, cancel terms, rewrite roots as powers, use identities (e.g. trig).
  2. Look for a derivative-of-inside pattern: f(g(x))g'(x) suggests substitution.
  3. Look for a product where one part simplifies when differentiated: suggests parts.
  4. If it’s rational: try polynomial division then partial fractions.
  5. If it’s definite (physics): set limits carefully, track units, and do a quick sanity check.

A. Substitution (u-sub)

If you see a pattern like f(g(x))g'(x), let u = g(x).

Template:

  1. Choose u = g(x).
  2. Compute du = g'(x) dx and rewrite dx.
  3. Change the whole integral into u-language, integrate, then substitute back.

Classic form:

∫ f'(x)/f(x) dx = ln |f(x)| + C.

B. Integration by parts

Use when you see a product where differentiating one part simplifies it:

∫ u dv = uv-∫ v du.

Template:

  1. Choose u so that du is simpler.
  2. Choose dv so you can integrate it to get v.
  3. Apply ∫ u dv = uv-∫ v du.

C. Partial fractions

Use for rational functions (polynomials divided by polynomials).

Example pattern:

1/(x²-1) = 1/((x-1)(x + 1)).
Not every integral has an elementary form

Some integrals involve special functions (error function, gamma function, etc.). In physics you usually approximate or use known standard results.

Pitfalls (especially in physics)
  • Missing +C on indefinite integrals.
  • Wrong limits on definite integrals (or forgetting to change limits after substitution).
  • Sign mistakes in work/potential integrals (e.g. Δ V = -∫ vector E · d vector ℓ).
  • Units: your final answer should have the correct physical units (a fast error detector).

3) Worked examples

Example 1: substitution

Compute:

∫ 2x/(x² + 1) dx.

Let u = x² + 1, so du = 2x dx:

∫ 2x/(x² + 1) dx = ∫ 1/u du = ln |u| + C = ln(x² + 1) + C.

Example 2: integration by parts

Compute:

∫ x e^x dx.

Choose u = x and dv = e^x dx. Then du = dx and v = e^x. So:

∫ x e^x dx = x e^x - ∫ e^x dx = x e^x - e^x + C = e^x(x-1) + C.

Example 3: partial fractions

Compute:

∫ 1/(x²-1) dx.

Decompose:

1/(x²-1) = A/(x-1) + B/(x + 1).

Solving gives A = 1/2 and B = -1/2, so:

∫ 1/(x²-1) dx = 1/2∫(1/(x-1)-1/(x + 1))dx = 1/2 ln |(x-1)/(x + 1)| + C.

Example 4: a physics-definite integral (RMS idea)

Compute:

∫₀^T cos² (ω t) dt.

Use cos² θ = (1/2)(1 + cos 2θ):

∫₀^T cos² (ω t) dt = 1/2∫₀^T 1 dt + 1/2∫₀^T cos(2ω t) dt = T/2 + (sin(2ω T))/4ω.

If T is a full number of periods (T = (2π n)/ω), then sin(2ω T) = 0 and:

∫₀^T cos² (ω t) dt = T/2.

Example 5 (physics): work done by a spring

For a Hooke’s-law spring, F(x) = kx (along the direction of motion). The work done stretching from x = 0 to x = X is:

W = ∫₀^X F(x) dx = ∫₀^X kx dx = 1/2 kX².

This is one of the most common “physics definite integrals”: set the limits from the physical endpoints.

Practice (with solutions)

1) Compute ∫ (3x^2) / (1 + x^3) dx

Let u = 1 + x³, so du = 3x² dx:

∫ 3x²/(1 + x³) dx = ∫ 1/u du = ln |u| + C = ln(1 + x³) + C.
2) Compute ∫ x cos x dx

Use parts: take u = x, dv = cos x dx so du = dx and v = sin x:

∫ x cos x dx = x sin x-∫ sin x dx = x sin x + cos x + C.
3) (Physics) A capacitor discharges: i(t)=I0 e^{-t/tau}. Find q(t)-q(0) from 0 to t

Hint: q(t)-q(0) = ∫₀^t i(t')dt'.

Answer:

q(t)-q(0) = ∫₀^t I₀e^(-t'/τ)dt' = I₀[-τ e^(-t'/τ)]₀^t = I₀τ(1-e^(-t/τ)).
4) (Physics sign) If E_x = k/x^2 along +x, compute V(b)-V(a)

Hint: V_b-Vₐ = -∫ₐ^b Eₓ dx.

Answer:

V(b)-V(a) = -∫ₐ^b (k/x²)dx = -[-k/x]ₐ^b = k(1/b-1/a).

  • Integration Table

    Use as a lookup once you can spot the patterns.

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  • Integration without integration

    A shortcut for e^ax sin(bx) and e^ax cos(bx) patterns.

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  • Differentiation Techniques

    Integration is reverse differentiation (plus constants).

    Open lesson

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