Explained: "Identify a technique of integration for evaluating the..."

nicekikah

Answered question

2021-10-19

Identify a technique of integration for evaluating the following integrals. If necessary, explain how to first simplify the integrand before applying the suggested technique of integration. You do not need to evaluate the integrals.

\int \frac{\tan^{2} x + 1}{\tan x} d x

Answer & Explanation

wheezym

Skilled2021-10-20Added 103 answers

Given that:

\int \frac{\tan^{2} x + 1}{\tan x} d x

Solving the integral through substitution.

\int \frac{\tan^{2} x + 1}{\tan x} = \int \frac{\sec^{2} x - 1 + 1}{\tan x}

\Rightarrow \int \frac{\tan^{2} x + 1}{\tan x} = \int \frac{\sec^{2} x}{\tan x}

Substitute

\tan x = t

\Rightarrow \sec^{2} x d x = d t

\Rightarrow \int \frac{\tan^{2} x + 1}{\tan x} = \int \frac{\sec^{2} x}{t} \times \frac{d t}{\sec^{2} x}

\Rightarrow \int \frac{\tan^{2} x + 1}{\tan x} = \int \frac{d t}{t}

\Rightarrow \int \frac{\tan^{2} x + 1}{\tan x} = \ln t + C

\Rightarrow \int \frac{\tan^{2} x + 1}{\tan x} = \ln | \tan x | + C

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