Expert answers to "Evaluate the integral. ∫x+1x2+2xdx"

High School
Calculus and Analysis
Calculus 2
Applications of integrals

lugreget9

Answered question

2021-12-28

Evaluate the integral.

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

Answer & Explanation

aquariump9

Beginner2021-12-29Added 40 answers

Step 1
Given: $I = \int \frac{x + 1}{x^{2} + 2 x} d x$
for evaluating given integral, we substitute
$x^{2} + 2 x = t$ ...(1)
now, differentiating equation (1) with respect to x
$\frac{d}{d x} (x^{2} + 2 x) = \frac{d}{d x} (t)$
$\frac{d}{d x} (x^{2}) + 2 \frac{d}{d x} (x) = \frac{d t}{d x} (∵ \frac{d}{d x} (x^{n}) = n x^{n - 1})$
$2 x + 2 (1) = \frac{d t}{d x}$
$2 x + 2 = \frac{d t}{d x}$
$2 (x + 1) = \frac{d t}{d x}$
$(x + 1) d x = \frac{d t}{2}$
Step 2
now, replacing (x+1)dx with $\frac{d t}{2}, (x^{2} + 2 x)$ with t in given integral
so,
$\int \frac{x + 1}{x^{2} + 2 x} d x = \frac{1}{2} \int \frac{d t}{t} (∵ \int \frac{d t}{t} = \ln | t | + c)$
$= \frac{1}{2} \ln | t | + c$ ...(2)
now replacing t with $(x^{2} + 2 x)$ in equation (2)
so,
$\int \frac{x + 1}{x^{2} + 2 x} d x = \frac{1}{2} \ln | x^{2} + 2 x | + c$
hence, given integral is equal to $\frac{1}{2} \ln | x^{2} + 2 x | + c$ .

porschomcl

Beginner2021-12-30Added 28 answers

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

Lets

Vasquez

Expert2022-01-07Added 669 answers

$\int \frac{x + 1}{x^{2} + 2 x} d x$
Transform
$\int \frac{1}{2 t} d t$
Use properties
$\frac{1}{2} \times \int \frac{1}{t} d t$
$\frac{1}{2} * \ln (| t |)$
Substitute back
$\frac{1}{2} * \ln (| x^{2} + 2 x |)$
Add C
Answer:
$\frac{1}{2} * \ln (| x^{2} + 2 x |) + C$

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