Integration Made Easy (Short Version) - Calculus

Discover the power rule for integration: ∫ x^n dx = x^{n+1}/(n+1) + C for n ≠ -1, with linearity and examples like x^2 and x^5.

Apply the power rule to integrate x^n for any n except -1, extend to negative powers with absolute values, and derive 1/x as ln|x| + C for all x.

Learn how to integrate exponential functions, starting with e^x whose integral equals itself. Use the constant rule for 1 and the power rule for x^n, and extend to other bases.

Discover basic integration rules for trigonometric functions, deriving antiderivatives for sine, cosine, sec^2, and tangent, and learn to rewrite expressions to fit the standard formulas.

Master a generalized integration formula for functions with inverse trig results, specifically ∫ dx/(x^2 + a^2) = (1/a) arctan(x/a) + C, with rewriting techniques.

Master integrating functions whose antiderivatives involve inverse trigonometric functions, applying formulas such as arcsin and arcsec forms with examples like 1/sqrt(a^2 - x^2) and 1/(x*sqrt(x^2 - a^2)).

Apply the u-substitution method to rewrite a complex integral in terms of u, choose u = x^2+1 with du = 2x dx, then use the power rule and substitute back.

Practice selecting the right u in u-substitution for integrals in polynomials, roots, and exponentials. Pause to guess u, apply derivative relationships, and allow constant factors when the derivative isn’t exact.