Standard Integrals of Particular Forms

Identify the exact form. Rewrite constants as squares. Check whether the expression is x^2+a^2, a^2-x^2, x^2-a^2, or a^2+x^2. Apply the matching formula. Include absolute value in logarithmic forms where required. Add +C for indefinite integrals.

A diagram is generally not needed. Formula recognition, sign checking, and constant-square rewriting are the useful supports.

For ∫dx/(x^2-9), writing (1/3)tan^(-1)(x/3)+C instead of the logarithmic form because the minus sign was ignored.

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.

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. Key forms for a>0: ∫dx/(x^2+a^2)=(1/a)tan^(-1)(x/a)+C; ∫dx/√(a^2-x^2)=sin^(-1)(x/a)+C; ∫dx/(x^2-a^2)=(1/(2a))ln|(x-a)/(x+a)|+C; ∫dx/(a^2-x^2)=(1/(2a))ln|(a+x)/(a-x)|+C. Conditions and signs must be checked. ∫dx/(x^2+a^2)=(1/a)tan^(-1)(x/a)+C for a>0.

These forms are solved by recognizing the exact structure and matching it with the correct standard formula. The sign between terms and the square-root placement decide whether the result involves tan inverse, sine inverse, logarithm, or another standard expression. Key forms for a>0: ∫dx/(x^2+a^2)=(1/a)tan^(-1)(x/a)+C; ∫dx/√(a^2-x^2)=sin^(-1)(x/a)+C; ∫dx/(x^2-a^2)=(1/(2a))ln|(x-a)/(x+a)|+C; ∫dx/(a^2-x^2)=(1/(2a))ln|(a+x)/(a-x)|+C. Conditions and signs must be checked. Evaluate ∫dx/(x^2+25). Here x^2+25=x^2+5^2. Using ∫dx/(x^2+a^2)=(1/a)tan^(-1)(x/a)+C, the answer is (1/5)tan^(-1)(x/5)+C. Often asked as direct standard-form questions or after completing the square. May also appear inside substitution-based integrals where the transformed integral matches a standard form. For ∫dx/(x^2-9), writing (1/3)tan^(-1)(x/3)+C instead of the logarithmic form because the minus sign was ignored.