7–64. Integration review Evaluate the following integrals.
38. ∫ x / (x⁴ + 2x² + 1) dx

Verified step by step guidance

1. Step 1: Observe the integrand ∫ x / (x⁴ + 2x² + 1) dx. Notice that the denominator can be factored. Rewrite the denominator as (x² + 1)², since x⁴ + 2x² + 1 is a perfect square trinomial.
2. Step 2: Substitute u = x² + 1 to simplify the integral. Compute the derivative of u with respect to x: du/dx = 2x, which implies du = 2x dx.
3. Step 3: Rewrite the integral in terms of u. Substitute x dx with (1/2) du, and the denominator becomes u². The integral now becomes (1/2) ∫ 1/u² du.
4. Step 4: Apply the power rule for integration to ∫ 1/u² du. Recall that ∫ u⁻² du = -u⁻¹ + C, where C is the constant of integration.
5. Step 5: Substitute back u = x² + 1 into the result to express the solution in terms of x. The final answer will be in the form of -1/(x² + 1) + C.

Key Concepts

Integration Techniques

Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and partial fraction decomposition. Understanding these methods is crucial for evaluating more complex integrals, such as the one presented in the question.

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers. In the integral ∫ x / (x⁴ + 2x² + 1) dx, the denominator is a polynomial of degree four. Recognizing the structure of polynomial functions helps in simplifying the integral and determining appropriate integration techniques.

Rational Functions

Rational functions are ratios of two polynomial functions. The integral in the question involves a rational function, which can often be simplified or decomposed for easier integration. Understanding how to manipulate rational functions is essential for effectively evaluating integrals like the one given.