Integrals Mind Map

If F'(x)=f(x), then the indefinite integral of f(x) is written as ∫f(x) dx=F(x)+C, where C is an arbitrary constant.

For every indefinite integral, check your answer by differentiating it. If the derivative gives the integrand, the integration is correct.

Integration by substitution changes a difficult integral into a standard one by putting u=g(x), so that du=g'(x) dx matches part of the integrand.

Look for an inside function whose derivative is also present. If the derivative is present with a constant multiplier, substitution is likely efficient.

Some trigonometric integrals are first simplified using identities, such as sin^2x=(1-cos2x)/2 and cos^2x=(1+cos2x)/2, before applying standard integration formulas.

Before integrating a trigonometric power or product, ask whether an identity can reduce it to first powers or a standard derivative pair.

Integrals of particular forms are standard results for expressions such as 1/(x^2+a^2), 1/(x^2-a^2), 1/√(a^2-x^2), and related algebraic forms.

Match the denominator exactly before applying a formula. Rewrite constants as squares whenever possible, such as x^2+9=x^2+3^2.

Integration by partial fractions decomposes a proper rational function into simpler fractions whose integrals are standard, usually logarithmic or inverse trigonometric.

After finding constants in the decomposition, recombine mentally or by substitution of easy x-values to check before integrating.

Integration by parts uses the formula ∫u dv=uv-∫v du to integrate a product of two functions, with u chosen so that its derivative becomes simpler.

Use ILATE as a guide, but still check whether differentiating u simplifies the expression and integrating dv is easy.

If F is an antiderivative of f on [a,b], then ∫_a^b f(x) dx=F(b)-F(a). This result is the evaluation form of the Fundamental Theorem of Calculus.

Always write the antiderivative in bracket notation before substituting limits; it reduces sign and order errors.

Properties of definite integrals transform the limits or integrand to simplify evaluation without first finding a complicated antiderivative.

For limits 0 to a, try replacing x by a-x. For limits -a to a, check whether the function is even or odd.