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Chapter 6: Problem 11

Find or evaluate the integral. $$ \int \tan ^{-1} x d x $$

Short Answer

Expert verified
The short version of the answer for the integral is: \[ \int \tan^{-1}(x) dx = x \tan^{-1}(x) - \frac{1}{2} \ln |1 + x^2| + C \]

Step by step solution

01

Select u and dv

Let u = \(\tan^{-1}(x)\) and dv = 1 dx. Now, we'll find the derivative and integral of these two functions:
02

Calculate du

To find du, we will differentiate u with respect to x: du \( = \frac{1}{1 + x^2} dx \)
03

Calculate v

To find v, we will integrate dv with respect to x: v \( = \int dv = \int 1 dx = x \) Now, we'll apply the integration by parts formula: \[ \int u \, dv = uv - \int v \, du \] with u, v, and du from above.
04

Apply the formula

Substituting u, v, and du into the formula, we get: \[ \int \tan^{-1}(x) dx = x \tan^{-1}(x) - \int x \frac{1}{1 + x^2} dx \] The unknown integral in this equation is a simple rational function, which can be solved using substitution.
05

Perform substitution

Let's make a substitution: Let \(y = 1 + x^2 \Rightarrow dy = 2x dx\). Then, we can rewrite the remaining unknown integral as: \[ \frac{1}{2} \int \frac{dy}{y} \] Now, we can find the integral.
06

Compute the unknown integral

Integrate with respect to y: \[ \frac{1}{2} \int \frac{dy}{y} = \frac{1}{2} (\ln |y| + C) \] Now, we need to substitute the original value of y back into this equation.
07

Back substitute and write the final answer

Substituting the value of y back into the equation, we have: \[ \frac{1}{2} (\ln |1 + x^2| + C) \] Recall that the result of the unknown integral is part of the integration by parts formula. Thus, substituting its value back into the formula gives the final answer: \[ \int \tan^{-1}(x) dx = x \tan^{-1}(x) - \frac{1}{2} \ln |1 + x^2| + C \] Here, C is the constant of integration.

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Most popular questions from this chapter

Find or evaluate the integral. $$ \int \frac{d x}{x^{4}+x^{2}+1} $$

Find or evaluate the integral. $$ \int \frac{\sqrt{x^{2}+9}}{x} d x $$

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. If \(f(x) \leq g(x)\) for all \(x\) in \([a, \infty)\) and \(\int_{a}^{\infty} f(x) d x\) converges, then \(\int_{a}^{\infty} g(x) d x\) also converges.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. Suppose that \(f\) is continuous on \([a, b)\) and \(f\) has an infinite discontinuity at \(b\). Furthermore, suppose that \(\int_{c}^{b} f(x) d x\) is convergent, where \(c\) is a number between \(a\) and \(b\). Then \(\int_{a}^{b} f(x) d x\) is convergent.

Find or evaluate the integral. \(\int \frac{t^{3}}{\sqrt{1-t^{2}}} d t\)
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