Calculate The Integral 4xln X3 Dx

This guide explains how to calculate the integral of 4xln(x³) dx using integration techniques. We'll cover the step-by-step process, provide a working calculator, and discuss common pitfalls when solving integrals.

How to Calculate the Integral

The integral of 4xln(x³) dx can be solved using integration by parts, which is a technique for finding the integral of a product of two functions. The formula for integration by parts is:

Integration by Parts Formula

∫u dv = uv - ∫v du

For our integral, we'll let:

u = ln(x³)
dv = 4x dx

Following the integration by parts process:

Find du by differentiating u: du = (1/x³)(3x²) dx = 3x dx
Find v by integrating dv: v = ∫4x dx = 2x² + C
Apply the integration by parts formula: ∫4xln(x³) dx = uv - ∫v du = ln(x³)(2x²) - ∫(2x²)(3x) dx
Simplify the remaining integral: ∫6x³ dx = (6/4)x⁴ = (3/2)x⁴
Combine terms: ∫4xln(x³) dx = 2x²ln(x³) - (3/2)x⁴ + C

Integration Techniques

When solving integrals, several techniques are commonly used:

Integration by Parts

Useful for integrals of products of functions. The LIATE rule (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) helps choose u and dv.

Substitution Method

Useful when the integrand contains a composite function. Let u = g(x) and express the integral in terms of u.

Partial Fractions

Used for rational functions where the numerator's degree is less than the denominator's.

Tip

Always simplify the integrand before attempting integration. For example, ln(x³) = 3ln(x), which can simplify the integral significantly.

Worked Example

Let's calculate ∫₀¹ 4xln(x³) dx using the formula we derived:

Final Integral Formula

∫4xln(x³) dx = 2x²ln(x³) - (3/2)x⁴ + C

Applying the limits from 0 to 1:

Evaluate at x=1: 2(1)²ln(1³) - (3/2)(1)⁴ = 0 - 3/2 = -3/2
Evaluate at x=0: 2(0)²ln(0³) - (3/2)(0)⁴ = 0 - 0 = 0
Subtract lower limit from upper limit: -3/2 - 0 = -3/2

The definite integral from 0 to 1 is -3/2.

Common Mistakes

When solving integrals, several common errors occur:

Incorrect Application of Integration by Parts

Choosing the wrong u and dv can lead to incorrect results. Always verify du and dv by differentiating and integrating.

Forgetting to Add the Constant of Integration

The constant C is essential for indefinite integrals. Omitting it means the solution is incomplete.

Miscounting Powers and Coefficients

When differentiating or integrating, it's easy to miscount powers or coefficients. Double-check each step.

Frequently Asked Questions

What is the integral of 4xln(x³) dx?: The integral is 2x²ln(x³) - (3/2)x⁴ + C.
How do I solve integrals with logarithmic functions?: Use integration by parts, letting u be the logarithmic function and dv be the remaining part of the integrand.
What is the difference between definite and indefinite integrals?: An indefinite integral includes a constant of integration (+C) and represents a family of functions. A definite integral has specific limits and yields a numerical value.
Can I use a calculator to solve integrals?: Yes, our interactive calculator can help you solve integrals quickly and accurately.
Where can I learn more about integration techniques?: Check out resources from MIT OpenCourseWare or Khan Academy for comprehensive integration tutorials.