Question

Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals. $$\int_{0}^{\pi / 4} \frac{\sin \theta}{\cos ^{3} \theta} d \theta$$

Short answer

Question: Evaluate the definite integral $$\int_{0}^{\pi / 4} \frac{\sin \theta}{\cos ^{3} \theta} d \theta$$ using a change of variables or a table of integrals. Answer: The definite integral is equal to $$-\frac{1}{2}$$

Step-by-step solution

Choose a suitable substitution

Let us use the substitution $$u = \cos \theta$$ which implies that $$du = -\sin \theta\, d\theta$$. Now we need to change the limits of integration according to the new variable \(u\).

Change the limits of integration

When \(\theta = 0\), we have: $$u = \cos 0 = 1$$ When \(\theta = \frac{\pi}{4}\), we have: $$u = \cos \frac{\pi}{4} = \frac{1}{\sqrt{2}}$$ So, the new limits of integration are \(u = 1\) and \(u = \frac{1}{\sqrt{2}}\).

Substitute and rewrite the integral

Now we substitute the relationship we found between \(\theta\) and \(u\), and also change the differential: $$\int_{0}^{\pi / 4} \frac{\sin \theta}{\cos ^{3} \theta} d \theta = -\int_{1}^{1/\sqrt{2}} \frac{1}{u^{3}} du$$

Evaluate the integral

We can now evaluate the integral: $$-\int_{1}^{1/\sqrt{2}} \frac{1}{u^{3}} du = -\left[ \frac{1}{2}u^{-2} \right]_{1}^{1/\sqrt{2}}$$

Calculate the definite integral

Now we need to substitute the limits of integration back in: $$- \left[\frac{1}{2}(1/\sqrt{2})^{-2} - \frac{1}{2}(1)^{-2}\right] = - \left[\frac{1}{2}(2) - \frac{1}{2}(1)\right] = - \frac{1}{2}$$ So the definite integral is equal to: $$\int_{0}^{\pi / 4} \frac{\sin \theta}{\cos ^{3} \theta} d \theta = -\frac{1}{2}$$

Key concepts

U-Substitution

U-substitution is a method used in calculus to simplify the process of integration, especially when the integral contains a function and its derivative. Think of it like a puzzle where you're trying to fit pieces together; once you identify a part of the integrand (the function to integrate) that can be replaced with a single variable, say 'u', the integral often becomes much easier to handle.

In our example, the substitution chosen is \( u = \text{cos} \, \theta \), which simplifies the integrand because the derivative of \( \text{cos} \, \theta \), which is \( -\text{sin} \, \theta \), is already present. With u-substitution, we're not only replacing the function with 'u' but also the differential (in this case, 'dθ') with 'du'. This transforms the integration process into a simpler one that is more straightforward to compute.

Trigonometric Integration

Trigonometric integration refers to the integration of functions involving trigonometric functions like sine, cosine, tangent, etc. These integrals can be tricky due to the periodic nature of trigonometric functions, and they require specific techniques or identities to solve.

The given exercise features a trigonometric function in the integrand, \( \frac{\text{sin} \, \theta}{\text{cos}^3 \, \theta} \). In such cases, one might use trigonometric identities to simplify the function or employ strategies such as u-substitution, where a trigonometric function is replaced by a variable 'u' to take advantage of the relationship between trigonometric functions and their derivatives.

Integration by Substitution

Integration by substitution is a powerful tool that, similar to u-substitution, helps to simplify integrals. This technique involves changing variables to transform the original integral into one that is easier to evaluate.

The crux of this method is to find a substitution that reduces the complexity of the original integral. After substituting the chosen function with 'u', and replacing 'dx' with 'du', the integrand's form changes. This can convert a seemingly complicated integral into a much simpler form, often to a basic integral that can be evaluated using fundamental integration rules.

Limits of Integration

The limits of integration define the interval over which we are integrating. When we employ a substitution such as u-substitution, it's imperative not only to change the integrand but also to adjust the limits of integration accordingly.

In our solved example, initially, we are integrating from 0 to \( \frac{\pi}{4} \). However, after the substitution \( u = \text{cos} \, \theta \), the limits are changed to 1 and \( \frac{1}{\sqrt{2}} \) to match the new variable 'u'. This step ensures that we are integrating over the correct interval in the 'u' variable. Neglecting to alter the limits accordingly would result in evaluating the integral over an incorrect range, leading to an erroneous result.