Integrals - Exam-Day Cram Sheet (Class 12)

Integrals - Exam-Day Cram Sheet

  • This comprehensive study guide covers Chapter 7 of Class 12 Mathematics, titled "Integrals," based on the NCERT curriculum. This revision sheet is designed as a quick, 10-minute read for exam preparation.

13 Standard Forms (Table 7.1)

  • Power Function Integral: The integral of a power of xx is given by the formula:

    • ∫xn dx=xn+1n+1+C\int x^n \, dx = \frac{x^{n+1}}{n+1} + C
    • Condition: This form is valid provided that n≠−1n \neq -1.

  • Logarithmic Integral: The integral of the reciprocal of xx is:

    • ∫dxx=log⁡∣x∣+C\int \frac{dx}{x} = \log |x| + C

  • Natural Exponential Integral: The integral of the exponential function with base ee is:

    • ∫ex dx=ex+C\int e^x \, dx = e^x + C

  • General Exponential Integral: The integral of an exponential function with base aa (aa being a positive constant) is:

    • ∫ax dx=axlog⁡a+C\int a^x \, dx = \frac{a^x}{\log a} + C

  • Cosine Integral: The integral of the cosine function is:

    • ∫cos⁡(x) dx=sin⁡(x)+C\int \cos(x) \, dx = \sin(x) + C

  • Sine Integral: The integral of the sine function is:

    • ∫sin⁡(x) dx=−cos⁡(x)+C\int \sin(x) \, dx = -\cos(x) + C

  • Secant Squared Integral: The integral of the square of the secant function is:

    • ∫sec⁡2(x) dx=tan⁡(x)+C\int \sec^2(x) \, dx = \tan(x) + C

  • Cosecant Squared Integral: The integral of the square of the cosecant function is:

    • ∫csc⁡2(x) dx=−cot⁡(x)+C\int \csc^2(x) \, dx = -\cot(x) + C

  • Secant-Tangent Integral: The integral of the product of secant and tangent is:

    • ∫sec⁡(x)tan⁡(x) dx=sec⁡(x)+C\int \sec(x) \tan(x) \, dx = \sec(x) + C

  • Cosecant-Cotangent Integral: The integral of the product of cosecant and cotangent is:

    • ∫csc⁡(x)cot⁡(x) dx=−csc⁡(x)+C\int \csc(x) \cot(x) \, dx = -\csc(x) + C

  • Inverse Sine Derivative Integral: The integral resulting in the inverse sine function is:

    • ∫dx1−x2=sin⁡−1(x)+C\int \frac{dx}{\sqrt{1-x^2}} = \sin^{-1}(x) + C

  • Inverse Tangent Derivative Integral: The integral resulting in the inverse tangent function is:

    • ∫dx1+x2=tan⁡−1(x)+C\int \frac{dx}{1+x^2} = \tan^{-1}(x) + C

Sign Tips for Integration

  • The "co-" Rule: A specific "Sign tip" is provided to identify which trigonometric results acquire a negative sign. Results involving "co-" functions (specifically cos, cot, and cosec) give a "mir" [interpreted as a minus sign based on context]. This matches patterns where:
    • ∫sin⁡(x) dx\int \sin(x) \, dx results in −cos⁡(x)-\cos(x).
    • ∫csc⁡2(x) dx\int \csc^2(x) \, dx results in −cot⁡(x)-\cot(x).
    • ∫csc⁡(x)cot⁡(x) dx\int \csc(x) \cot(x) \, dx results in −csc⁡(x)-\csc(x).

Six Special Forms (Section 7.4)

  • Integration of the Form intfracdxx2−a2\\int \\frac{dx}{x^2 - a^2}:
    • The result for this special rational form is:
    • ∫dxx2−a2=12alog⁡∣x−ax+a∣+C\int \frac{dx}{x^2 - a^2} = \frac{1}{2a} \log \left| \frac{x-a}{x+a} \right| + C