Integration by Parts Calculator

Integration by Parts Calculator

Use this calculator to organize the components of the integration by parts formula: ∫u dv = uv – ∫v du. Enter your chosen functions and differentials, and the calculator will display the assembled formula.

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Understanding Integration by Parts

Integration by Parts is a fundamental technique in calculus used to integrate products of functions. It is derived from the product rule for differentiation. The formula states:

∫u dv = uv – ∫v du

Where:

• u is a function of x that becomes simpler when differentiated.
• dv is a differential that can be easily integrated to find v.
• du is the differential of u (i.e., the derivative of u multiplied by dx).
• v is the integral of dv.

How to Choose 'u' and 'dv' (LIATE Rule)

The most crucial step in integration by parts is correctly choosing which part of the integrand will be u and which will be dv. A common mnemonic to help with this choice is LIATE:

1. Logarithmic functions (e.g., ln(x))
2. Inverse trigonometric functions (e.g., arctan(x))
3. Algebraic functions (e.g., x, x²)
4. Trigonometric functions (e.g., sin(x), cos(x))
5. Exponential functions (e.g., e^x)

You generally choose u to be the function that appears earliest in the LIATE list. The remaining part of the integrand becomes dv.

Steps to Apply Integration by Parts:

1. Identify u and dv: Based on the LIATE rule, select u and the remaining part as dv.
2. Calculate du: Differentiate u to find du.
3. Calculate v: Integrate dv to find v. (Remember to include the constant of integration only at the very end of the entire problem, not for intermediate v).
4. Apply the Formula: Substitute u, v, du, and dv into the formula: ∫u dv = uv - ∫v du.
5. Solve the Remaining Integral: The new integral ∫v du should ideally be simpler to solve than the original integral. If not, you might need to re-evaluate your choice of u and dv, or apply integration by parts again.

Example Usage with the Calculator:

Let's integrate ∫x e^x dx.

1. Choose u and dv:
   • According to LIATE, 'x' is Algebraic (A) and 'e^x' is Exponential (E). 'A' comes before 'E'.
   • So, let u = x.
   • The remaining part is dv = e^x dx.
2. Calculate du:
   • Differentiate u = x: du/dx = 1, so du = dx.
3. Calculate v:
   • Integrate dv = e^x dx: v = ∫e^x dx = e^x.
4. Enter into Calculator:
   • Function u: x
   • Differential dv: e^x dx
   • Differential du: dx
   • Function v: e^x
5. Click "Calculate Formula". The calculator will display:

∫ (x) (e^x dx) = (x)(e^x) – ∫ (e^x) (dx)

From here, you would solve the remaining integral ∫e^x dx, which is e^x. So the final answer is x e^x – e^x + C.