What's 4-2k=-5(k+7 this is to hard for and I don't get this at all

Find the integral using substitution or a formula.

Nadusha1986 [10]

$\rm \int \dfrac{x^2+7}{x^2+2x+5}~dx$

Derivative of the denominator:
$\rm (x^2+2x+5)'=2x+2$

Hmm our numerator is 2x+7. Ok this let's us know that a simple u-substitution is NOT going to work. But let's apply some clever Algebra to the numerator splitting it up into two separate fractions. Split the +7 into +2 and +5.

$\rm \int \dfrac{x^2+2+5}{x^2+2x+5}~dx$

and then split the fraction,

$\rm \int \dfrac{x^2+2}{x^2+2x+5}~dx+\int\dfrac{5}{x^2+2x+5}~dx$

Based on our previous test, we know that a simple substitution will work for the first integral: $\rm \quad u=x^2+2x+5\qquad\to\qquad du=2x+2~dx$

So the first integral changes,

$\rm \int \dfrac{1}{u}~du+\int\dfrac{5}{x^2+2x+5}~dx$

integrating to a log,

$\rm ln|x^2+2x+5|+\int\dfrac{5}{x^2+2x+5}~dx$

Other one is a little tricky. We'll need to complete the square on the denominator. After that it will look very similar to our arctangent integral so perhaps we can just match it up to the identity.

$\rm x^2+2x+5=(x^2+2x+1)+4=(x+1)^2+2^2$

So we have this going on,

$\rm ln|x^2+2x+5|+\int\dfrac{5}{(x+1)^2+2^2}~dx$

Let's factor the 5 out of the intergral,
and the 4 from the denominator,

$\rm ln|x^2+2x+5|+\frac54\int\dfrac{1}{\frac{(x+1)^2}{2^2}+1}~dx$

Bringing all that stuff together as a single square,

$\rm ln|x^2+2x+5|+\frac54\int\dfrac{1}{\left(\dfrac{x+1}{2}\right)^2+1}~dx$

Making the substitution: $\rm \quad u=\dfrac{x+1}{2}\qquad\to\qquad 2du=dx$

giving us,

$\rm ln|x^2+2x+5|+\frac54\int\dfrac{1}{\left(u\right)^2+1}~2du$

simplying a lil bit,

$\rm ln|x^2+2x+5|+\frac52\int\dfrac{1}{u^2+1}~du$

and hopefully from this point you recognize your arctangent integral,

$\rm ln|x^2+2x+5|+\frac52arctan(u)$

undo your substitution as a final step,
and include a constant of integration,

$\rm ln|x^2+2x+5|+\frac52arctan\left(\frac{x+1}{2}\right)+c$

Hope that helps!
Lemme know if any steps were too confusing.

8 0

3 years ago

What is the measure of ∠4? Enter your answer in the box.

charle [14.2K]

Step-by-step explanation:

$\angle 4, \:60\degree \:\&\:61\degree$ are straight line angles.

$\therefore \angle \: 4 + 60 \degree + 60 \degree = 180\degree \\ \\ \therefore \: \angle \: 4 = 180\degree - 121\degree \\ \\ \huge \red {\boxed {\therefore \: \angle \: 4 = 59\degree}}$

4 0

3 years ago

Read 2 more answers

Mrs taylor gave each of her students 75 pieces paper at the beginning of the school year if there are 32 students in her class h

Jobisdone [24]

Problem
Mrs. Taylor handed out 75 sheets of paper to each student in her class. If she has 32 students, how many total sheets of paper did she hand out?

Result
2,400 sheets of paper

Solution
We can solve this using multiplication.

75 · 32 = 2,400

3 0

3 years ago

Read 2 more answers

Lee is staying at a hotel the cost of the first night is 240$ the cost for each night after that is 210$ let y represent the tot

alukav5142 [94]

Let's write the function that represents the problem:
ax = 210x + 240
We are going to evaluate the first five nights:
a0 = 210 * (0) + 240 = 240
a1 = 210 * (1) + 240 = 450
a2 = 210 * (2) + 240 = 660
a3 = 210 * (3) + 240 = 870
a4 = 210 * (3) + 240 = 1080
We then have the following terms:
240, 450, 660, 870, 1080
We observe that the common difference of the terms is:
210
Therefore, since they have the same common difference, we are facing an arithmetic sequence.
Answer:
A the situation represents an arithmetic sequence because of the successive and values have a common difference of 210

7 0

3 years ago

Suppose you take a class from each of the following four professors: 1, 2, 3, and 4. You write five papers and get the following

harina [27]

Answer:

Professor 1 as he gave the maximum number of A's .

5 0

3 years ago