Math Problem Statement

3dx /(x(x2 + 4)2)

Solution

The given expression is:

3dxx(x2+4)2\int \frac{3 \, dx}{x(x^2 + 4)^2}

We are tasked with finding the indefinite integral of this function.

Step-by-step Solution:

  1. Substitute u=x2+4u = x^2 + 4, which implies du=2xdxdu = 2x \, dx. So, xdx=du2x \, dx = \frac{du}{2}.

  2. The integral becomes: 3dxx(x2+4)2=32duu2\int \frac{3 \, dx}{x(x^2 + 4)^2} = \frac{3}{2} \int \frac{du}{u^2}

  3. The integral of 1u2\frac{1}{u^2} is 1u-\frac{1}{u}.

  4. Substituting back u=x2+4u = x^2 + 4, we get: 321x2+4-\frac{3}{2} \cdot \frac{1}{x^2 + 4}

  5. Therefore, the indefinite integral is: 32(x2+4)+C-\frac{3}{2(x^2 + 4)} + C where CC is the constant of integration.

Would you like more details on any specific part of this solution?

Related Questions:

  1. How would you approach a similar integral with a different denominator, say x2+ax^2 + a?
  2. What changes if the power of (x2+4)(x^2 + 4) was different, like 3 or higher?
  3. Can we solve this integral using partial fractions?
  4. How can we compute the definite integral of this expression over a specific range?
  5. What is the geometric interpretation of the integral 1x2+4\frac{1}{x^2 + 4}?

Tip:

When solving integrals with quadratic terms, trigonometric substitution often simplifies the process, especially when the quadratic expression resembles a trigonometric identity like 1+tan2(θ)=sec2(θ)1 + \tan^2(\theta) = \sec^2(\theta).

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Math Problem Analysis

Mathematical Concepts

Integration
Substitution Method
Indefinite Integrals

Formulas

Integral of 1/u^n is -1/(n-1) * u^(n-1) for n ≠ 1
Substitution: u = x^2 + 4, du = 2x dx

Theorems

Substitution Rule for Integration

Suitable Grade Level

Undergraduate Calculus