Problem 7 - Calculus Early Transcendentals

Question

How would you evaluate \(\int \tan ^{10} x \sec ^{2} x d x ?\)

Step-by-Step Solution

Answer

Question: Evaluate the integral \(\int \tan^{10} x \sec^2 x d x\). Answer: \(\frac{(\tan x)^{11}}{11} + C\)

Step 1: Identify the substitution variables

To proceed with substitution, we'll set \(u = \tan x\). Now we need to find the derivative of \(u\) with respect to \(x\). We know that the derivative of \(\tan x\) with respect to \(x\) is \(\sec^2 x\). Therefore, we have \(d u = \sec^2 x d x\).

Step 2: Rewrite the integral using the substitution variables

Now we can rewrite the given integral using the substitution variables we determined in step 1: \(\int \tan^{10} x \sec^2 x d x = \int u^{10} d u\)

Step 3: Integrate the new expression

Now we have to integrate \(u^{10}\) with respect to \(u\). This is a straightforward power rule integration: \(\int u^{10} d u = \frac{u^{11}}{11} + C\)

Step 4: Substitute back the variable x

The final step is to substitute back the original variable "\(x\)" in place of "\(u\)". Since we know \(u = \tan x\): \(\frac{u^{11}}{11} + C = \frac{(\tan x)^{11}}{11} + C\) So the integral of \(\tan^{10} x \sec^2 x d x\) is \(\frac{(\tan x)^{11}}{11} + C\).