Question

Evaluate the following integrals. $$\int \sec ^{2} x \tan ^{1 / 2} x d x$$

Short answer

Question: Evaluate the integral \(\int \sec ^{2} x \tan ^{1 / 2} x d x\). Answer: \(\frac{2}{3}(\tan x)^{3/2} + C\)

Step-by-step solution

Perform substitution

Let \(u = \tan x\). Then we have \(u^2 = \tan^2 x\) and \(\sec^2 x = 1 + \tan^2 x = 1 + u^2\). To find \(du\), we differentiate \(u\) with respect to \(x\): $$\frac{d u}{d x} = \frac{d}{d x}(\tan x) = \sec ^{2}x$$ Now we can solve for \(dx\): $$d x = \frac{d u}{\sec ^{2}x}$$ Now, we can substitute \(u\) and \(dx\) back into the integral: $$\int \sec ^{2} x \tan ^{1 / 2} x d x = \int (1+u^2) u^{1/2} \frac{d u}{\sec ^{2}x}$$

Simplify the integral and integrate

Notice that the factor \(\sec^2 x\) in the numerator and denominator cancel each other out. Also, substitute the expression for \(\sec^2 x\) in terms of \(u\) back into the integral: $$\int (1+u^2) u^{1/2} \frac{d u}{1 + u^2} = \int u^{1/2} du$$ Now, we can integrate with respect to \(u\): $$\int u^{1/2} du = \frac{2}{3}u^{3/2} + C$$

Substitute back in terms of \(x\)

Finally, we need to substitute \(x\) back into our result. Since we let \(u = \tan x\), we have: $$\frac{2}{3}u^{3/2} + C = \frac{2}{3}(\tan x)^{3/2} + C$$ Thus, the final result is: $$\int \sec ^{2} x \tan ^{1 / 2} x d x = \frac{2}{3}(\tan x)^{3/2} + C$$

Key concepts

Substitution Method

The substitution method is a powerful tool in calculus, particularly in integration. It helps transform a complicated integral into a simpler one by substituting part of the integral with a new variable.

• Identify a substitution that will simplify the integral.
• Replace the selected variable with a new variable, usually denoted as u.
• Find the derivative of u with respect to x to determine du.
• Write dx in terms of du.
• Rewrite the entire integral in terms of u, eliminating the original variable x.

In the exercise above, we selected the substitution of u=tan(x). This choice simplifies the integral significantly, as its derivative with respect to x is sec²(x), which matches perfectly with the structure of the integrand. By transforming the integral into the variable u, you reduce the complexity and make it easier to solve.

Trigonometric Integrals

Trigonometric integrals involve the integration of products of trigonometric functions. These integrals often require specific techniques such as substitution or trigonometric identities to be solvable. It is essential to understand the properties and relationships between trigonometric functions for these kinds of integrals.

• Identify trigonometric identities that can simplify parts of the integrand, such as \( \sec^2 x = 1 + \tan^2 x \).
• Use substitution to change the trigonometric expression into a simpler form, if applicable.

In the given exercise, recognizing that \( \sec^2 x = 1 + \tan^2 x \) allowed us to effectively perform a substitution and simplify the integral. This approach showcases the importance of both trigonometric identities and substitution in solving integrals involving trigonometric functions.

Algebraic Manipulation

Algebraic manipulation is fundamental for simplifying expressions before carrying out integration. It involves rearranging, factoring, or expanding expressions in a way that makes the integration process more manageable.

• Cancel out common terms to simplify the expression.
• Use factorization techniques, if necessary, to further simplify integrands.
• Expand or condense terms to fit standard integral forms.

In the exercise, once the substitution was made, we noticed that \( \sec^2 x \) in the integrand's numerator and denominator canceled each other out, simplifying the problem. This shows how algebraic manipulation can play a critical role in reducing a complex integral into an easily solvable one, highlighting the elegance and effectiveness of integrating algebraic skills into calculus methods.