Mathematics, 30.01.2020 14:59 braydenmcd02

Don't answer if you don't know the answer.

find the indefinite integral

Answers: 2

Show answers

Answers

Answer from: gabriellebarnes66

$\displaystyle \int {\frac{x + 2}{x^2 + 1}} \, dx = 2arctan(x) + \frac{ln|x^2 + 1|}{2} + C$

General Formulas and Concepts:

Algebra I

Terms/CoefficientsExpanding

Calculus

Differentiation

DerivativesDerivative Notation

Derivative Property [Addition/Subtraction]: $\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]$

Basic Power Rule:

f(x) = cxⁿf’(x) = c·nxⁿ⁻¹

Integration

Integrals[Indefinite Integrals] Integration Constant C

Integration Rule [Reverse Power Rule]: $\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C$

Integration Property [Multiplied Constant]: $\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx$

Integration Property [Addition/Subtraction]: $\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx$

U-Substitution

Step-by-step explanation:

Step 1: Define

Identify

$\displaystyle \int {\frac{x + 2}{x^2 + 1}} \, dx$

Step 2: Integrate Pt. 1

Set variables for u-substitution.

Set u: $\displaystyle u = x^2 + 1$ [u] Differentiate [Basic Power Rule, Addition/Subtraction]: $\displaystyle du = 2x \ dx$

Step 3: Integrate Pt. 2

[Integral] Rewrite [Integration Property - Multiplied Constant]: $\displaystyle \int {\frac{x + 2}{x^2 + 1}} \, dx = \frac{1}{2}\int {\frac{2(x + 2)}{x^2 + 1}} \, dx$ [Integrand] Expand: $\displaystyle \int {\frac{x + 2}{x^2 + 1}} \, dx = \frac{1}{2}\int {\frac{2x + 4}{x^2 + 1}} \, dx$ [Integral] Rewrite [Integration Property - Addition/Subtraction]: $\displaystyle \int {\frac{x + 2}{x^2 + 1}} \, dx = \frac{1}{2} \bigg[ \int {\frac{2x}{x^2 + 1}} \, dx + \int {\frac{4}{x^2 + 1}} \, dx \bigg]$ [2nd Integral] Rewrite [Integration Property - Multiplied Constant]: $\displaystyle \int {\frac{x + 2}{x^2 + 1}} \, dx = \frac{1}{2} \bigg[ \int {\frac{2x}{x^2 + 1}} \, dx + 4\int {\frac{1}{x^2 + 1}} \, dx \bigg]$ [1st Integral] U-Substitution: $\displaystyle \int {\frac{x + 2}{x^2 + 1}} \, dx = \frac{1}{2} \bigg[ \int {\frac{1}{u}} \, du + 4\int {\frac{1}{x^2 + 1}} \, dx \bigg]$ [1st Integral] Logarithmic Integration: $\displaystyle \int {\frac{x + 2}{x^2 + 1}} \, dx = \frac{1}{2} \bigg[ ln|u| + 4\int {\frac{1}{x^2 + 1}} \, dx \bigg]$ [Integral] Arctrig Integration: $\displaystyle \int {\frac{x + 2}{x^2 + 1}} \, dx = \frac{1}{2} \bigg[ ln|u| + 4 \bigg( \frac{1}{1}arctan \Big( \frac{x}{1} \Big) \bigg) \bigg] + C$ Simplify: $\displaystyle \int {\frac{x + 2}{x^2 + 1}} \, dx = \frac{1}{2} \bigg[ ln|u| + 4arctan(x) \bigg] + C$ Expand: $\displaystyle \int {\frac{x + 2}{x^2 + 1}} \, dx = 2arctan(x) + \frac{ln|u|}{2} + C$ Back-Substitute: $\displaystyle \int {\frac{x + 2}{x^2 + 1}} \, dx = 2arctan(x) + \frac{ln|x^2 + 1|}{2} + C$

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Integration

Answer from: Quest

Use a calculator basically

Answer from: Quest

If there are four friends then each would get, 1.25 or 1 1/4. so this is how i solved 5 ÷ 4 = 1.25 . hope this ; )

Other questions on the subject: Mathematics

Mathematics, 21.06.2019 15:30, xxxamslashxxx9

Data was collected on myrtle beach for 11 consecutive days. each day the temperature and number of visitors was noted. the scatter plot below represents this data. how many people visited the beach when the temperature was 84 degrees?

Answers: 1

continue

Mathematics, 21.06.2019 23:00, BeautyxQueen

Who long does it take to drive 150 miles at 45 miles per hour

Answers: 2

continue

Mathematics, 22.06.2019 00:20, Jaymiegrimes22

Traffic on saturday, it took ms. torres 24 minutes to drive 20 miles from her home to her office. during friday's rush hour, it took 75 minutes to drive the same distance. a. what was ms. torres's average speed in miles per hour on saturday? b. what was her average speed in miles per hour on friday?

Answers: 1

continue

Mathematics, 22.06.2019 01:00, jadensababe9151

On friday, there were x students at the baseball game. on monday, there were half as many students at the game as there were on friday. on wednesday, there were 32 fewer students at the game as there were on friday. which expression could represent the total number of tickets sold for all 3 games?

Answers: 2

continue

You know the right answer?

Don't answer if you don't know the answer.

find the indefinite integral...