Integral Calculator

The fundamental rules for finding antiderivatives.

Constant Rule

∫k dx = kx + C

The integral of a constant is that constant times x.

• ∫5 dx = 5x + C
• ∫(-3) dx = -3x + C

Power Rule

∫x^n dx = x^(n+1)/(n+1) + C (n ≠ -1)

The most important integration rule.

• ∫x² dx = x³/3 + C
• ∫x^4 dx = x^5/5 + C
• ∫√x dx = ∫x^(1/2) dx = (2/3)x^(3/2) + C
• ∫(1/x²) dx = ∫x^(-2) dx = -1/x + C

Special Case: n = -1

∫(1/x) dx = ln|x| + C

Cannot use power rule when n = -1.

Constant Multiple Rule

∫c·f(x) dx = c·∫f(x) dx

Constants factor out of integrals.

• ∫3x² dx = 3·∫x² dx = 3·(x³/3) + C = x³ + C
• ∫5x^4 dx = 5·(x^5/5) + C = x^5 + C

Sum/Difference Rule

∫[f(x) ± g(x)] dx = ∫f(x) dx ± ∫g(x) dx

Integrate term by term.

• ∫(x² + 3x) dx = x³/3 + 3x²/2 + C
• ∫(x³ - x + 1) dx = x^4/4 - x²/2 + x + C

Integrals of the basic trigonometric functions.

Basic Trig Integrals

With Chain Rule Adjustment

When integrating trig(ax): ∫trig(ax) dx = (1/a)·∫trig(u) du where u = ax

Examples:

• ∫sin(3x) dx = -(1/3)cos(3x) + C
• ∫cos(2x) dx = (1/2)sin(2x) + C
• ∫sec²(5x) dx = (1/5)tan(5x) + C

Verification

Always verify by differentiating:

• d/dx[-(1/3)cos(3x)] = -(1/3)·(-sin(3x))·3 = sin(3x) ✓

Common Trig Identities for Integration

Sometimes rewriting helps:

• sin²(x) = (1 - cos(2x))/2
• cos²(x) = (1 + cos(2x))/2
• tan²(x) = sec²(x) - 1

Integrals involving e^x and logarithms.

Natural Exponential

∫e^x dx = e^x + C

The exponential function is its own antiderivative!

With chain rule: ∫e^(ax) dx = (1/a)e^(ax) + C

Examples:

• ∫e^(2x) dx = (1/2)e^(2x) + C
• ∫e^(-x) dx = -e^(-x) + C
• ∫3e^(4x) dx = (3/4)e^(4x) + C

General Exponential

∫a^x dx = a^x/ln(a) + C

Examples:

• ∫2^x dx = 2^x/ln(2) + C
• ∫10^x dx = 10^x/ln(10) + C

Logarithmic Functions

∫ln(x) dx = x·ln(x) - x + C (integration by parts)

∫(1/x) dx = ln|x| + C

Combining with Other Rules

Example: ∫x·e^x dx (requires integration by parts)

Let u = x, dv = e^x dx Then du = dx, v = e^x

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

Definite Integrals

∫[0,1] e^x dx = e^1 - e^0 = e - 1 ≈ 1.718

∫[1,e] (1/x) dx = ln(e) - ln(1) = 1 - 0 = 1

Computing definite integrals and their geometric meaning.

The Fundamental Theorem

If F'(x) = f(x), then: ∫[a,b] f(x) dx = F(b) - F(a)

Computing Definite Integrals

Step 1: Find the indefinite integral (antiderivative) Step 2: Evaluate at upper bound Step 3: Evaluate at lower bound Step 4: Subtract: F(b) - F(a)

Example Calculation

∫[1,3] x² dx

1. Antiderivative: F(x) = x³/3
2. F(3) = 27/3 = 9
3. F(1) = 1/3
4. Result: 9 - 1/3 = 26/3 ≈ 8.667

Geometric Interpretation

The definite integral gives the signed area:

• Positive where f(x) > 0 (above x-axis)
• Negative where f(x) < 0 (below x-axis)

Properties of Definite Integrals

Reversal: ∫[a,b] f(x) dx = -∫[b,a] f(x) dx

Zero width: ∫[a,a] f(x) dx = 0

Additivity: ∫[a,c] f(x) dx = ∫[a,b] f(x) dx + ∫[b,c] f(x) dx

Comparison: If f(x) ≥ g(x) on [a,b], then ∫[a,b] f(x) dx ≥ ∫[a,b] g(x) dx

U-substitution: the reverse of the chain rule.

The Method

If the integral has the form ∫f(g(x))·g'(x) dx:

1. Let u = g(x)
2. Then du = g'(x) dx
3. Substitute: ∫f(u) du
4. Integrate
5. Substitute back x

Example 1: ∫2x·cos(x²) dx

Let u = x², then du = 2x dx

∫2x·cos(x²) dx = ∫cos(u) du = sin(u) + C = sin(x²) + C

Example 2: ∫x²·√(x³+1) dx

Let u = x³ + 1, then du = 3x² dx, so x² dx = du/3

∫x²·√(x³+1) dx = (1/3)∫√u du = (1/3)·(2/3)u^(3/2) + C = (2/9)(x³+1)^(3/2) + C

Example 3: ∫(1/(2x+3)) dx

Let u = 2x + 3, then du = 2 dx, so dx = du/2

∫(1/(2x+3)) dx = (1/2)∫(1/u) du = (1/2)ln|u| + C = (1/2)ln|2x+3| + C

When to Use Substitution

Look for:

• A function and its derivative present
• Composite functions f(g(x))
• Chain rule patterns
• Expressions like (ax + b)^n

The product rule in reverse for integrating products.

The Formula

∫u dv = uv - ∫v du

From the product rule: d(uv) = u dv + v du

LIATE Rule for Choosing u

Choose u in this priority order:

• Logarithmic (ln x, log x)
• Inverse trig (arcsin, arctan)
• Algebraic (x, x², polynomials)
• Trigonometric (sin, cos)
• Exponential (e^x)

Example 1: ∫x·e^x dx

u = x (algebraic), dv = e^x dx du = dx, v = e^x

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

Example 2: ∫x²·sin(x) dx

u = x², dv = sin(x) dx du = 2x dx, v = -cos(x)

∫x²·sin(x) dx = -x²cos(x) + 2∫x·cos(x) dx

(Apply by parts again to remaining integral)

Example 3: ∫ln(x) dx

u = ln(x), dv = dx du = (1/x) dx, v = x

∫ln(x) dx = x·ln(x) - ∫x·(1/x) dx = x·ln(x) - x + C

Tabular Method

For repeated integration by parts (like ∫x³·e^x dx), use the tabular method to organize derivatives of u and integrals of dv.