To solve an integral, you first determine if it's an indefinite integral (finding the antiderivative) or a definite integral (finding the area under a curve between two points). Indefinite integrals involve using techniques like u-substitution, integration by parts, or trigonometric substitution, while definite integrals also require the Fundamental Theorem of Calculus.
1. Identify the Type of Integral:

Indefinite Integral:
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No limits of integration are specified. The goal is to find a general antiderivative, which will include a constant of integration (+C).
Definite Integral:
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Limits of integration (upper and lower bounds) are specified. The goal is to find a numerical value representing the area under the curve between those bounds.

2. Indefinite Integrals: Finding the Antiderivative

Basic Power Rule:
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For integrals of the form ∫xⁿ dx, the antiderivative is (xⁿ⁺¹)/(n+1) + C, where n ≠ -1.
U-Substitution:
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If a composite function is present, make a substitution (like u = g(x)) to simplify the integral and then perform the substitution back.
Integration by Parts:
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Used for integrals of the form ∫u dv, where u is a function of x and v is the integral of a different function. The formula is uv - ∫v du.
Trigonometric Substitution:
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When square roots or other trigonometric functions are present, trigonometric substitutions can help simplify the integral.

3. Definite Integrals: Applying the Fundamental Theorem of Calculus

Evaluate the antiderivative at the upper limit of integration.
Evaluate the antiderivative at the lower limit of integration.
Subtract the lower limit value from the upper limit value.

4. Examples:

Indefinite Integral: ∫2x³ dx = (2x⁴)/4 + C = x⁴/2 + C

Definite Integral: ∫₀¹ x² dx = [x³/3]₀¹ = (1³/3) - (0³/3) = 1/3