Problem 23

Question

Find the indefinite integral. $$ \int \frac{\sec x \tan x}{\sec x-1} d x $$

Step-by-Step Solution

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Answer

The indefinite integral of the given function is $ ln |sec x - 1| + C $

1Step 1: Determine Suitable Substitution

Let's do a substitution: let $u = sec x -1$. Then differential of $u$ is given by $du = sec x \cdot tan x dx $. This substitution simplifies the integrand to $ \int \frac{du}{u} $

2Step 2: Solve the Integral

The integral now reduces to the form that we know easily: $ \int \frac{du}{u} = ln |u| + C $

3Step 3: Substitute back original variable

Substitute back the original variable in place of $u$ to get the final answer as $ln |sec x - 1| + C $

Problem 23

Other exercises in this chapter

Problem 23

Evaluate the definite integral of the transcendental function. Use a graphing utility to verify your result. $$ \int_{-\pi / 3}^{\pi / 3} 4 \sec \theta \tan \th

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Problem 23

Solve the differential equation. $$ \frac{d y}{d x}=4 x+\frac{4 x}{\sqrt{16-x^{2}}} $$

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Problem 23

Find the indefinite integral and check the result by differentiation. $$ \int\left(2 \sin x-5 e^{x}\right) d x $$

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Problem 24

Find the derivative of the function. $g(x)=\operatorname{sech}^{2} 3 x$

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