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Integration by substitution is a method used to simplify the process of integration. The idea is to make a substitution to turn a complicated integral into a simpler one by replacing a variable with a new one. Imagine you are faced with the integral \(\int_{0}^{1 / 2} x^{4}(1-2 x)^{3} \mathrm{~d} x\). First, identify a substitution that will simplify the integral. Here, let \(\ u = 1 - 2x \), so when \(\ x = 0, u = 1 \) and when \(\ x = 1/2, u = 0 \). Differentiating \(\ u \) with respect to \(\ x \), we get \(\ du = -2 dx \). So the integral becomes \(\int_{1}^{0} \frac{-1}{2} u^{3}(u/2) \mathrm{~d} u \). After rewriting the limits, switch them and adjust the sign to get \(\int_{0}^{1} \frac{1}{4} u^{4} \mathrm{~d} u \). This simplifies further, and you can find the result by ordinary integration.

Integration by parts is a technique based on the product rule for differentiation. It is useful when integrating the product of two functions. The formula is \(\ \int u \mathrm{~d}v = uv - \int v \mathrm{~d}u \) where \(\ u \) and \(\ v \) are functions of \(\ x \). Suppose we have \(\int x e^{x} \mathrm{~d}x \). Let \(\ u = x \), and \(\ dv = e^{x} \mathrm{~d}x \). Then, \(\ du = \mathrm{~d}x \) and \(\ v = e^{x} \). Using integration by parts, we get \(\ x e^{x} - \int e^{x} \mathrm{~d}x = x e^{x} - e^{x} + C \), which simplifies to \(\ e^{x}(x - 1) + C \). This method is especially useful for trigonometric and exponential integrals.

Trigonometric integrals involve integrating trigonometric functions. This often requires using trigonometric identities to simplify the integral. For example, consider \(\ \int_{0}^{\pi / 2} \sin^{5} \theta \cos^{4} \theta \mathrm{~d} \theta \). A useful identity here is \(\ \sin^{2} \theta + \cos^{2} \theta = 1 \). Break down \(\sin^{5} \theta \) as \(\sin^{4} \theta \sin \theta = (\sin^{2} \theta)^{2} \sin \theta = (1 - \cos^{2} \theta)^{2} \sin \theta \). This transforms the integral into a polynomial of \(\cos \theta \) times \(\sin \theta \), which is more straightforward to integrate.

When you make a substitution in a definite integral, you also need to change the limits of integration to match the new variable. For instance, if evaluating \(\int_{0}^{1 / \sqrt{2}} x^{2} \sqrt{1-2 x^{2}} \mathrm{~d} x \) you let \(\ u = 1 - 2x^{2} \). The limits change as follows: when \(\ x = 0, u = 1 \) and when \(\ x = 1/ \sqrt{2}, u = 0 \). This changes the bounds of the integral, allowing us to integrate with respect to \(\ u \), transforming the integral to \(\int_{0}^{1} \frac{1}{-4} \sqrt{u} \mathrm{~d} u \). Remember to adjust the integrand accordingly and evaluate within the new limits.

Evaluating definite integrals involves computing the integral and then applying the limits. For example, consider the integral \(\ \int_{0}^{1/3} x^{2} \sqrt{1-9 x^{2}} \mathrm{~d} x \). Use substitution \(\ u = 1 - 9x^{2} \), giving \(\ du = -18x \mathrm{~d} x \). Change the limits: when \(\ x = 0 \), \(\ u = 1 \), and when \(\ x = 1/3 \), \(\ u = 0 \). Rewrite the integral as \(\int_{1}^{0} \frac{-1}{18} u^{1/2} \mathrm{~d} u \). Flipping the limits and simplifying, it becomes \(\int_{0}^{1} \frac{1}{18} u^{1/2} \mathrm{~d} u \). Integrate to find the antiderivative and then evaluate from 0 to 1 to obtain the final result.