23-64. Integration Evaluate the following integrals.
41. ∫₋₁¹ x/(x + 3)² dx

Verified step by step guidance

1. First, observe the integral \( \int_{-1}^{1} \frac{x}{(x + 3)^2} \, dx \). Notice that the integrand is a rational function where the denominator is \( (x+3)^2 \).
2. To simplify the integral, consider using substitution. Let \( u = x + 3 \). Then, \( du = dx \) and \( x = u - 3 \). Also, change the limits of integration accordingly: when \( x = -1 \), \( u = 2 \), and when \( x = 1 \), \( u = 4 \).
3. Rewrite the integral in terms of \( u \): \( \int_{2}^{4} \frac{u - 3}{u^2} \, du \). This can be separated into two simpler integrals: \( \int_{2}^{4} \frac{u}{u^2} \, du - 3 \int_{2}^{4} \frac{1}{u^2} \, du \).
4. Simplify the integrands: \( \frac{u}{u^2} = \frac{1}{u} \) and \( \frac{1}{u^2} = u^{-2} \). So the integral becomes \( \int_{2}^{4} \frac{1}{u} \, du - 3 \int_{2}^{4} u^{-2} \, du \).
5. Now, integrate each term separately: \( \int \frac{1}{u} \, du = \ln|u| \) and \( \int u^{-2} \, du = -u^{-1} \). After integrating, apply the limits \( 2 \) to \( 4 \) to find the definite integral.

Key Concepts

Definite Integration

Definite integration calculates the net area under a curve between two limits. It involves evaluating the integral of a function from a lower to an upper bound, resulting in a numerical value representing accumulated quantity or area.

Integration by Substitution

Integration by substitution simplifies integrals by changing variables to transform the integral into a more manageable form. It is useful when the integrand contains a composite function, allowing easier integration after substitution.

Handling Rational Functions

Rational functions are ratios of polynomials, often requiring algebraic manipulation or substitution for integration. Recognizing patterns like derivatives of denominators in the numerator helps in applying substitution or partial fraction techniques.