Complete Integral Formula Sheet: All Essential Indefinite and Definite Integral Formulas

1. Basic Integral Rules

∫ 0 dx = C
∫ k dx = kx + C
∫ x dx = x²/2 + C
∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + C     (n ≠ -1)
∫ 1/x dx = ln|x| + C

2. Exponential and Logarithmic Integrals

∫ eˣ dx = eˣ + C
∫ aˣ dx = aˣ / ln(a) + C
∫ ln(x) dx = x ln(x) − x + C
∫ e^(kx) dx = e^(kx) / k + C

3. Trigonometric Integrals

∫ sin(x) dx = −cos(x) + C
∫ cos(x) dx = sin(x) + C
∫ tan(x) dx = −ln|cos(x)| + C
∫ cot(x) dx = ln|sin(x)| + C
∫ sec(x) dx = ln|sec(x) + tan(x)| + C
∫ csc(x) dx = ln|csc(x) − cot(x)| + C

4. Trigonometric Power Integrals

∫ sin²(x) dx = (x/2) − (sin(2x)/4) + C
∫ cos²(x) dx = (x/2) + (sin(2x)/4) + C
∫ sec²(x) dx = tan(x) + C
∫ csc²(x) dx = −cot(x) + C

5. Inverse Trigonometric Integrals

∫ 1/√(1 − x²) dx = arcsin(x) + C
∫ −1/√(1 − x²) dx = arccos(x) + C
∫ 1/(1 + x²) dx = arctan(x) + C
∫ 1/(|x|√(x² − 1)) dx = arcsec(x) + C

6. Hyperbolic Function Integrals

∫ sinh(x) dx = cosh(x) + C
∫ cosh(x) dx = sinh(x) + C
∫ tanh(x) dx = ln|cosh(x)| + C
∫ sech²(x) dx = tanh(x) + C

7. Integration by Substitution

If u = g(x), then:

∫ f(g(x)) g'(x) dx = ∫ f(u) du

8. Integration by Parts

∫ u dv = uv − ∫ v du

9. Rational Function Integrals

∫ 1/(x² + a²) dx = (1/a) arctan(x/a) + C
∫ 1/(x² − a²) dx = (1/2a) ln|(x − a)/(x + a)| + C
∫ x/(x² + a²) dx = (1/2) ln(x² + a²) + C

10. Definite Integral Properties

∫aa f(x) dx = 0
∫ab f(x) dx = − ∫ba f(x) dx
∫ab [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx

11. Fundamental Theorem of Calculus

If F'(x) = f(x), then:

∫ab f(x) dx = F(b) − F(a)

12. Special Integrals

∫ √(a² − x²) dx = (x/2)√(a² − x²) + (a²/2) arcsin(x/a) + C
∫ √(x² − a²) dx = (x/2)√(x² − a²) − (a²/2) ln|x + √(x² − a²)| + C
∫ √(a² + x²) dx = (x/2)√(a² + x²) + (a²/2) ln|x + √(a² + x²)| + C

Conclusion

This formula sheet covers all essential integral formulas used in calculus, engineering, physics, and mathematics. With these rules, you can evaluate most standard integrals and build a strong foundation for advanced problem‑solving.