Integration Formulas

Constant: \(\int k dx = kx + C\)

Power Rule: \( \int x^n \,dx = \frac{{x^{n+1}}}{{n+1}} + C \), for \( n \neq -1 \)

Exponential: \( \int e^x \,dx = e^x + C \)

Natural Logarithm: \( \int \ln(x) \,dx = x \ln(x) - x + C \)

Trigonometric Functions:

• \( \int \sin(x) \,dx = -\cos(x) + C \)
• \( \int \cos(x) \,dx = \sin(x) + C \)
• \( \int \sec^2(x) \,dx = \tan(x) + C \)
• \( \int \csc^2(x) \,dx = -\cot(x) + C \)
• \( \int \sec(x) \tan(x) \,dx = \sec(x) + C \)
• \( \int \csc(x) \cot(x) \,dx = -\csc(x) + C \)

Inverse Trigonometric Functions:

• \( \int \frac{dx}{a^2 + x^2} = \frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right) + C \)
• \( \int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}\left(\frac{x}{a}\right) + C \)
• \( \int \frac{dx}{x\sqrt{x^2 - a^2}} = \frac{1}{a}\sec^{-1}\left(\frac{x}{a}\right) + C \)
• \( \int -\frac{dx}{x \sqrt{x^2 - a^2}} = \frac{1}{a}\csc^{-1}\left(\frac{x}{a}\right) + C \)

Hyperbolic Functions:

• \( \int \sinh(x) \,dx = \cosh(x) + C \)
• \( \int \cosh(x) \,dx = \sinh(x) + C \)
• \( \int {sech}^2(x) \,dx = -\tanh(x) + C \)
• \( \int {csch}^2(x) \,dx = -\coth(x) + C \)
• \( \int \sec(x) \tan(x) \,dx = -{sech}(x) + C \)
• \( \int \csc(x) \cot(x) \,dx = -{csch}(x) + C \)

Logarithmic Functions:

• \( \int \log_a(x) \,dx = \frac{x \log_a(x) - x}{\ln(a)} + C \)

Rational Functions: \( \int R(x) \,dx \), where \( R(x) \) is a rational function, may involve techniques like partial fractions decomposition.

Special Functions: Some integrals involving special functions like the error function, gamma function, Bessel functions, etc., may not have simple closed-form expressions.

Irrational Functions:

Q1 Find the integral of \( \int \frac{dx}{1-4x^2} \)

• Solution: To solve the given integral we need to use the formula of irrational integral \( \int \frac{dx}{1- x^2} = \sin^{-1}\left(x\right) + C \).