Question

Multiple substitutions If necessary, use two or more substitutions to find the following integrals. $$\int \tan ^{10} 4 x \sec ^{2} 4 x d x(\text {Hint}: \text { Begin with } u=4 x .)$$

Short answer

Question: Evaluate the integral $$\int \tan^{10} 4x \sec^2 4x dx$$ and simplify using two substitutions. Answer: $$\frac{1}{4} \left( \frac{\tan ^{13} 4x}{13} + \frac{\tan ^{11} 4x}{11} \right) + C$$

Step-by-step solution

Apply the first substitution \(u = 4x\)

Perform the substitution \(u = 4x\), which means \(du = 4dx\). So, \(dx = \frac{1}{4}du\). We substitute these new variables into the integral: $$ \int \tan^{10}(u) \sec^2(u) \cdot \frac{1}{4} du $$ Now we will simplify: $$ \frac{1}{4} \int \tan^{10}(u) \sec^2(u) du $$

Apply the Pythagorean trigonometric identity \(\sec^2 x = (\tan^2 x + 1)\)

Using the Pythagorean trigonometric identity, \(\sec^2 u = (\tan^2 u + 1)\): $$ \frac{1}{4} \int \tan^{10}(u)(\tan^2 (u) + 1) du $$ Now we distribute the \(\tan^{10}(u)\) inside the parentheses: $$ \frac{1}{4} \int (\tan^{12}(u) + \tan^{10}(u)) du $$

Apply the second substitution \(v = \tan u\)

Perform a second substitution \(v = \tan(u)\), which means \(dv = \sec^2(u) du\). We substitute these new variables into the integral: $$ \frac{1}{4} \int (v^{12} + v^{10}) dv $$ Now we will integrate the powers of \(v\) with respect to \(v\): $$ \frac{1}{4} \left[ \frac{v^{13}}{13} + \frac{v^{11}}{11} \right] + C $$

Substitute back for the original variable \(x\)

Now, we will replace \(v\) with \(\tan(u)\) and then replace \(u\) with \(4x\): $$ \frac{1}{4} \left[ \frac{\tan^{13}(4x)}{13} + \frac{\tan^{11}(4x)}{11} \right] + C $$ And this is our final answer: $$ \boxed{\frac{1}{4} \left( \frac{\tan ^{13} 4x}{13} + \frac{\tan ^{11} 4x}{11} \right) + C} $$

Key concepts

Pythagorean trigonometric identity

In trigonometry, the Pythagorean trigonometric identity is a fundamental concept. This identity states that for any angle \(x\), the equation \(\sec^2 x = 1 + \tan^2 x\) holds true. It originates from the Pythagorean theorem applied to a right triangle. This identity is particularly useful in calculus, especially when dealing with integrals involving trigonometric functions.

In the given exercise, we used this identity to manipulate and simplify the integral. By rewriting \(\sec^2(u)\) as \((\tan^2(u) + 1)\), we made it easier to distribute and integrate terms. Understanding this identity is essential for tackling trigonometric integrals because it frequently provides an avenue to transform complex integrals into simpler ones.

Here's the key point: whenever you encounter \(\sec^2 x\) in an integral, remember you can substitute it with \(1 + \tan^2 x\). This can help break down the integral into terms that are often easier to integrate directly.

Multiple substitutions

The method of multiple substitutions is a powerful technique in calculus, used to simplify complex integrals. The key is to strategically apply one substitution after another until the integral becomes manageable. Typically, each substitution aims to introduce a simpler variable or transforms the integral into a more familiar form.

In this problem, the first substitution we used was \(u = 4x\), which reduced the integral involving \(4x\) to a new variable \(u\). By changing variables, the derivative \(dx\) could be expressed in terms of \(du\), simplifying the expression. The second substitution \(v = \tan(u)\) was chosen because it directly relates to the trigonometric functions present. In this case, it turned \(\sec^2(u)\) into \(dv\), making integration straightforward.

This approach often requires careful planning and a good understanding of the relationships between trigonometric functions and their derivatives. When executed correctly, substitutions allow you to transform a daunting integral into something more familiar and manageable. Look out for repeating themes or functions similar to previously solved integrals, as they often provide clues for effective substitutions.

Trigonometric integrals

Trigonometric integrals are integrals involving trigonometric functions such as sine, cosine, tangent, and their combinations. These integrals often require specific strategies for evaluation, including substitution and the use of trigonometric identities.

The integral in the example \(\int \tan ^{10} 4 x \sec ^{2} 4 x \; dx\) showcases several techniques needed for solving trigonometric integrals. Techniques such as trigonometric identities and careful substitutions help in breaking down such expressions into simpler terms.

• **Understand Trigonometric Identities:** These are tools for rewriting trig expressions. For example, reducing \(\sec^2 x\) using \(\sec^2 x = 1 + \tan^2 x\) facilitated a substitution.
• **Use of Substitutions:** By mitigating one complexity at a time, you move step-by-step towards solving the integral.
• **Revert Back After Integration:** After integrating in terms of the new variables, remember to substitute back the original variable to express the solution in context.

If you're meeting these integrals for the first time, practice is key. With experience, substituting, simplifying, and solving trigonometric integrals will be more intuitive.