Check the solution to "Determine the following indefinite integral. ∫2+x21+x2dx"

High School
Calculus and Analysis
Calculus 2
Applications of integrals

jubateee

Answered question

2021-12-30

Determine the following indefinite integral.

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

Answer & Explanation

Foreckije

Beginner2021-12-31Added 32 answers

Step 1
Given: indefinite integral

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

To solve the indefinite integral, first simplify the function,

\frac{2 + x^{2}}{1 + x^{2}} = \frac{1 + x^{2} + 1}{1 + x^{2}}

= 1 + \frac{1}{1 + x^{2}}

Step 2
Substitute simplified function in the given integral.

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

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

= \tan^{- 1} x + x + c

Step 3
Check the solution by differentiation

\frac{d}{d x} (\arctan (x) + x + c) = \frac{d}{d x} (\arctan (x)) + \frac{d}{d x} (x) + \frac{d}{d x} (c)

= \frac{1}{x^{2} + 1} + 1 + 0

= \frac{1 + x^{2} + 1}{x^{2} + 1}

= \frac{2 + x^{2}}{x^{2} + 1}

Therefore, this matches with the given integral.

redhotdevil13l3

Beginner2022-01-01Added 30 answers

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

Rewrite the expression

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

Separate the fraction

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

Use properties of integrals

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

Evaluate the integrals

\arctan (x) + x

Add C
Solution

\arctan (x) + x + C

karton

Expert2022-01-04Added 613 answers

$\begin{matrix} \frac{x^{2} + 2}{x^{2} + 1} d x \\ = \int (\frac{x^{2} + 1}{x^{2} + 1} + \frac{1}{x^{2} + 1}) d x \\ = \int (\frac{1}{x^{2} + 1} + 1) d x \\ = \int \frac{1}{x^{2} + 1} d x + \int 1 d x \\ \int \frac{1}{x^{2} + 1} d x \\ = \arctan (x) \\ \int 1 d x \\ = x \\ \int \frac{1}{x^{2} + 1} d x + \int 1 d x \\ = \arctan (x) + x \\ \int \frac{x^{2} + 2}{x^{2} + 1} \\ = \arctan (x) + x + C \end{matrix}$

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