Integral Calculator - Complete Guide to Integration Methods

The power rule handles basic polynomials. But real integrals involve composite functions, products, and ugly fractions. That's where u-substitution, integration by parts, and partial fractions come in.

Here's when to use each method — with examples.

Method 1: Power Rule — The Foundation

∫xⁿ dx = xⁿ⁺¹/(n+1) + C

Use for: simple polynomials, single-term integrands.

Example: ∫(4x³ − 6x + 2) dx = x⁴ − 3x² + 2x + C

Method 2: U-Substitution — Chain Rule in Reverse

When the integrand has a function and its derivative, u-substitution simplifies it.

∫f(g(x)) · g'(x) dx → ∫f(u) du where u = g(x)

Example: ∫2x·(x²+1)⁴ dx

1. Let u = x² + 1 → du = 2x dx
2. Substitute: ∫u⁴ du = u⁵/5 + C
3. Back-substitute: (x²+1)⁵/5 + C

Example: ∫6x²·√(2x³+5) dx

1. Let u = 2x³ + 5 → du = 6x² dx
2. Substitute: ∫√u du = ∫u^(1/2) du = (2/3)u^(3/2) + C
3. Back-substitute: (2/3)(2x³+5)^(3/2) + C

When to use u-sub: You see a composite function AND its derivative's "pieces" in the integrand.

Method 3: Integration by Parts — Product Rule in Reverse

∫u dv = uv − ∫v du

Choose u and dv using the LIATE priority (pick u first from the leftmost available):

1. Logarithmic (ln x)
2. Inverse trig (arctan x)
3. Algebraic (x², 3x)
4. Trigonometric (sin x, cos x)
5. Exponential (eˣ)

Example: ∫x·eˣ dx

1. u = x (algebraic), dv = eˣ dx
2. du = dx, v = eˣ
3. ∫x·eˣ dx = x·eˣ − ∫eˣ dx = x·eˣ − eˣ + C

Example: ∫x²·sin(x) dx

1. u = x², dv = sin(x) dx
2. du = 2x dx, v = −cos(x)
3. = −x²·cos(x) + ∫2x·cos(x) dx
4. Apply by parts again to ∫x·cos(x) dx
5. = −x²·cos(x) + 2[x·sin(x) − ∫sin(x) dx]
6. = −x²·cos(x) + 2x·sin(x) + 2cos(x) + C

Method 4: Partial Fractions — Rational Functions

When integrating a rational function P(x)/Q(x) where Q factors nicely:

Example: ∫(3x+7)/((x+1)(x+2)) dx

1. Decompose: (3x+7)/((x+1)(x+2)) = A/(x+1) + B/(x+2)
2. 3x+7 = A(x+2) + B(x+1)
3. x = −1: 4 = A → A = 4
4. x = −2: 1 = −B → B = −1
5. ∫4/(x+1) dx − ∫1/(x+2) dx
6. = 4ln|x+1| − ln|x+2| + C

Which Method? Decision Tree

| Integrand Type | Method | Example | |---------------|--------|---------| | Simple polynomial | Power rule | ∫x⁴ dx | | Composite + derivative pieces | U-substitution | ∫2x(x²+1)³ dx | | Two different function types multiplied | Integration by parts | ∫xeˣ dx | | Rational function (polynomial/polynomial) | Partial fractions | ∫1/((x+1)(x−2)) dx |

The Trench Truth: On exams, 70% of non-trivial integrals use u-substitution. Always check for u-sub first — it's the fastest method. If you can't find a good u, then try integration by parts. Partial fractions come up less often but are worth easy points when they do.

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