Here are some basic integral formulas that are essential in calculus:

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

2. Integral of a Constant: $$ \int a , dx = ax + C $$

3. Sum Rule: $$ \int (f(x) + g(x)) , dx = \int f(x) , dx + \int g(x) , dx $$

4. Difference Rule: $$ \int (f(x) - g(x)) , dx = \int f(x) , dx - \int g(x) , dx $$

5. Constant Multiple Rule: $$ \int a \cdot f(x) , dx = a \int f(x) , dx $$

6. Integral of Exponential Functions: $$ \int e^x , dx = e^x + C $$ $$ \int a^x , dx = \frac{a^x}{\ln(a)} + C \quad \text{(for } a > 0 \text{ and } a \neq 1\text{)} $$

7. Integral of Trigonometric Functions: $$ \int \sin(x) , dx = -\cos(x) + C $$ $$ \int \cos(x) , dx = \sin(x) + C $$ $$ \int \sec^2(x) , dx = \tan(x) + C $$ $$ \int \csc^2(x) , dx = -\cot(x) + C $$ $$ \int \sec(x) \tan(x) , dx = \sec(x) + C $$ $$ \int \csc(x) \cot(x) , dx = -\csc(x) + C $$

8. Integral of Hyperbolic Functions: $$ \int \sinh(x) , dx = \cosh(x) + C $$ $$ \int \cosh(x) , dx = \sinh(x) + C $$

9. Integral of the Natural Logarithm: $$ \int \ln(x) , dx = x \ln(x) - x + C $$

These formulas form the foundation for solving a wide variety of integral problems.

Monash version

• $\int 1𝑑𝑥=𝑥+𝐶$
• $\int 𝑎𝑑𝑥=𝑎𝑥+𝐶$
• $\int 𝑥𝑛𝑑𝑥=((𝑥𝑛+1)/(𝑛+1))+𝐶;𝑛≠1$
• $\int sin(𝑥)𝑑𝑥=−cos(𝑥)+𝐶$
• $\int cos(𝑥)𝑑𝑥=sin(𝑥)+𝐶$
• $\int sec^2(𝑥)𝑑𝑥=tan(𝑥)+𝐶$
• $\int csc^2(𝑥)𝑑𝑥=−cot(𝑥)+𝐶$
• $\int sec(𝑥(tan(𝑥))𝑑𝑥=sec𝑥+𝐶$
• $\int csc𝑥(cot𝑥)𝑑𝑥=−csc𝑥+𝐶$
• $\int (1/𝑥)𝑑𝑥=ln|𝑥|+𝐶$
• $\int 𝑒𝑥𝑑𝑥=𝑒𝑥+𝐶$
• $\int 𝑎𝑥𝑑𝑥=(𝑎𝑥/ln𝑎)+𝐶;𝑎>0,𝑎≠1$

Sum Rule of Integration: $\int(𝑓+𝑔)𝑑𝑥=\int𝑓𝑑𝑥+\int𝑔𝑑𝑥$

Difference Rule of Integration: $\int (𝑓−𝑔)𝑑𝑥=\int 𝑓𝑑𝑥−\int 𝑔𝑑𝑥$

Multiplication by Constant $\int 𝑐𝑓(𝑥)𝑑𝑥=𝑐\int 𝑓(𝑥)𝑑𝑥$.