integration farmula

It seems like you're asking about integration formulas in mathematics. Integration involves finding the antiderivative of a function, which is essentially the reverse process of differentiation. Here are some common integration formulas:

1. **Power Rule**: ∫x^n dx = (1/(n+1)) * x^(n+1) + C, where n ≠ -1.

2. **Constant Rule**: ∫c dx = cx + C, where c is a constant.

3. **Exponential Rule**: ∫e^x dx = e^x + C.

4. **Trigonometric Integrals**:

   - ∫sin(x) dx = -cos(x) + C.

   - ∫cos(x) dx = sin(x) + C.

   - ∫sec^2(x) dx = tan(x) + C.

   - ∫csc^2(x) dx = -cot(x) + C.

5. **Inverse Trigonometric Integrals**:

   - ∫(1/√(1-x^2)) dx = arcsin(x) + C.

   - ∫(1/(1+x^2)) dx = arctan(x) + C.

6. **Integration by Parts**: ∫u dv = uv - ∫v du, where u and v are differentiable functions.

7. **Integration of Rational Functions**:

   - ∫(1/x) dx = ln|x| + C.

   - ∫(1/(x^2 + a^2)) dx = (1/a) arctan(x/a) + C.

8. **Integration of Exponential Functions**:

   - ∫a^x dx = (1/ln(a)) * a^x + C.

9. **Integration of Hyperbolic Functions**:

   - ∫sinh(x) dx = cosh(x) + C.

   - ∫cosh(x) dx = sinh(x) + C.

Remember that integration often involves adding an arbitrary constant of integration (usually denoted as "C") because the antiderivative is not unique – it can differ by a constant value.

These are just a few basic integration formulas. Depending on the complexity of the function you're integrating, you might need to use more advanced techniques like trigonometric substitutions, partial fractions, or numerical methods. 

Certainly! Here are some trigonometric integration formulas that are commonly used:

1. **Sine and Cosine Functions**:

   - ∫sin(ax) dx = -(1/a) * cos(ax) + C.

   - ∫cos(ax) dx = (1/a) * sin(ax) + C.

2. **Powers of Trigonometric Functions**:

   - ∫sin^n(x) dx = -(1/n) * sin^(n-1)(x) * cos(x) + (n-1)/n * ∫sin^(n-2)(x) dx.

   - ∫cos^n(x) dx = (1/n) * cos^(n-1)(x) * sin(x) + (n-1)/n * ∫cos^(n-2)(x) dx.

3. **Products of Trigonometric Functions**:

   - ∫sin(ax) cos(bx) dx = (1/(a+b)) * sin((a+b)x) + C.

   - ∫sin(ax) sin(bx) dx = -(1/(a-b)) * cos((a-b)x) + C.

   - ∫cos(ax) cos(bx) dx = (1/(a+b)) * sin((a+b)x) + C.

   4. **Trigonometric Substitutions**:

   Trigonometric substitutions involve using trigonometric identities to simplify integrals. For example:

   - ∫√(a^2 - x^2) dx, where a is a constant, can be simplified using the identity sin^2(θ) + cos^2(θ) = 1.

5. **Inverse Trigonometric Functions**:

   - ∫(1/√(a^2 - x^2)) dx = arcsin(x/a) + C.

   - ∫(1/√(x^2 - a^2)) dx = arccos(x/a) + C.

   - ∫(1/(x^2 + a^2)) dx = (1/a) arctan(x/a) + C.

6. **Tangent and Secant Functions**:

   - ∫tan(x) dx = -ln|cos(x)| + C.

   - ∫sec(x) dx = ln|sec(x) + tan(x)| + C.

These are just a few examples of trigonometric integration formulas. Depending on the specific problem, you might need to use trigonometric identities, substitutions, or other techniques to simplify the integrals before applying these formulas. Keep in mind that trigonometric integrals can sometimes be quite challenging, and creative approaches might be necessary for more complex cases.