Use a substitution to reduce the following integrals to ∫ ln u du. Then evaluate using the formula for ∫ ln x dx.
7. ∫ (sec²x) · ln(tan x + 2) dx

Verified step by step guidance

1. Identify a substitution that simplifies the integral. Notice that the argument of the logarithm is \( \tan x + 2 \), and the derivative of \( \tan x \) is \( \sec^2 x \), which appears as a factor in the integral. So, let \( u = \tan x + 2 \).
2. Compute the differential \( du \) in terms of \( dx \). Since \( u = \tan x + 2 \), then \( \frac{du}{dx} = \sec^2 x \), which implies \( du = \sec^2 x \, dx \).
3. Rewrite the integral in terms of \( u \) and \( du \). The integral \( \int (\sec^2 x) \cdot \ln(\tan x + 2) \, dx \) becomes \( \int \ln u \, du \).
4. Recall the formula for the integral of \( \ln x \): \( \int \ln x \, dx = x \ln x - x + C \). Use this formula to evaluate \( \int \ln u \, du \).
5. After integrating, substitute back \( u = \tan x + 2 \) to express the answer in terms of \( x \).

Key Concepts

Substitution Method in Integration

The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. It involves identifying a part of the integrand as a new variable u, then rewriting the integral in terms of u and du. This technique is especially useful when the integral contains a composite function.

Integral of ln x

The integral of the natural logarithm function, ∫ ln x dx, can be evaluated using integration by parts. The formula is ∫ ln x dx = x ln x - x + C. This result is essential when the integral reduces to a form involving ln u, allowing direct evaluation after substitution.

Derivative of tan x and sec² x

Recognizing that the derivative of tan x is sec² x is crucial for substitution in this problem. Since sec² x dx equals d(tan x), it allows the integral involving sec² x and ln(tan x + 2) to be rewritten in terms of u = tan x + 2, facilitating the reduction to ∫ ln u du.