integrate 1/(x)(x^2 + 5)
i can't put it as 1/(1/2*u) thn i bring the 1/(1/2) outside first? leaving me with integrating 1/u du?

Let's go step-by-step to solve the integral:

\[
\int \frac{1}{x(x^2 + 5)} \, dx
\]

### Step 1: Use substitution
We can use **partial fractions** to simplify the integral. Let's first decompose the fraction.

1. The expression \(\frac{1}{x(x^2 + 5)}\) can be split into:
   \[
   \frac{A}{x} + \frac{Bx + C}{x^2 + 5}
   \]
   Multiply both sides by \(x(x^2 + 5)\) to get:
   \[
   1 = A(x^2 + 5) + (Bx + C)x
   \]
   Expanding:
   \[
   1 = A(x^2) + 5A + Bx^2 + Cx
   \]
   Group the terms:
   \[
   1 = (A + B)x^2 + Cx + 5A
   \]
   Now, equate coefficients of powers of \(x\):
   - For \(x^2\): \(A + B = 0\)
   - For \(x^1\): \(C = 0\)
   - Constant: \(5A = 1\)

   From \(5A = 1\), we get:
   \[
   A = \frac{1}{5}
   \]

   From \(A + B = 0\), we get:
   \[
   B = -\frac{1}{5}
   \]

   And \(C = 0\).

   So, the partial fraction decomposition is:
   \[
   \frac{1}{x(x^2 + 5)} = \frac{1/5}{x} - \frac{1/5}{x^2 + 5}
   \]

### Step 2: Integrate the two terms separately

Now we can integrate each term:

\[
\int \frac{1}{x(x^2 + 5)} \, dx = \frac{1}{5} \int \frac{1}{x} \, dx - \frac{1}{5} \int \frac{1}{x^2 + 5} \, dx
\]

1. The first integral is straightforward:
   \[
   \frac{1}{5} \int \frac{1}{x} \, dx = \frac{1}{5} \ln |x|
   \]

2. The second integral:
   \[
   \int \frac{1}{x^2 + 5} \, dx
   \]
   This is a standard form, where \(\int \frac{1}{x^2 + a^2} \, dx = \frac{1}{a} \tan^{-1} \left( \frac{x}{a} \right)\).
   
   Here, \(a^2 = 5\), so \(a = \sqrt{5}\):
   \[
   \int \frac{1}{x^2 + 5} \, dx = \frac{1}{\sqrt{5}} \tan^{-1} \left( \frac{x}{\sqrt{5}} \right)
   \]

### Step 3: Final answer

Now substitute back into the original expression:

\[
\int \frac{1}{x(x^2 + 5)} \, dx = \frac{1}{5} \ln |x| - \frac{1}{5 \sqrt{5}} \tan^{-1} \left( \frac{x}{\sqrt{5}} \right) + C
\]

### Summary:
The integral is:

\[
\int \frac{1}{x(x^2 + 5)} \, dx = \frac{1}{5} \ln |x| - \frac{1}{5 \sqrt{5}} \tan^{-1} \left( \frac{x}{\sqrt{5}} \right) + C
\]

---

Would you like more details on any step or have any questions?

### 5 Related Questions:
1. How does partial fraction decomposition work in more complex cases?
2. Can we integrate this using substitution only instead of partial fractions?
3. What is the geometric interpretation of the \(\tan^{-1}\) term in integrals?
4. How do we handle integrals of higher degree polynomials in the denominator?
5. What other methods exist for solving rational function integrals?

### Tip:
When dealing with tricky integrals involving rational functions, always check if partial fractions can simplify the process.

integrate 1/(x)(x^2 + 5) i can't put it as 1/(1/2*u) thn i bring the 1/(1/2) outside first? leaving me with integrating 1/u du?

Learn how to integrate the function 1/(x(x^2 + 5)) with a detailed step-by-step solution. This calculus problem covers key concepts such as partial fraction decomposition and trigonometric substitution, applying the logarithmic and arctangent integration formulas. Suitable for undergraduate calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation