Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Integrate.
int(x+5)/(x^(2)-2x-3)dx

Integrate. $\newline$ $\int \frac{x+5}{x^{2}-2 x-3} d x$

View step-by-step help

View step-by-step help

Evaluate definite integrals using the power rule

Full solution

Q. Integrate.

\newline

\int \frac{x+5}{x^{2}-2 x-3} d x

Factor Denominator: Factor the denominator of the integrand. $\newline$ The denominator $x^2 - 2x - 3$ can be factored into $(x - 3)(x + 1)$ .
Partial Fraction Decomposition: Perform partial fraction decomposition. $\newline$ We want to express $(x+5)/(x^2-2x-3)$ as $A/(x-3) + B/(x+1)$ .
Find $A$ and $B$ : Find the values of $A$ and $B$ . To find $A$ and $B$ , we multiply both sides by the denominator $(x^2-2x-3)$ to get: $x + 5 = A(x + 1) + B(x - 3)$ . Now we can solve for $A$ and $B$ by choosing suitable values for $B$ $0$ .
Solve for $B$ : Choose $x = 3$ to solve for $B$ . $\newline$ Plugging $x = 3$ into the equation, we get: $\newline$ $3 + 5 = A(3 + 1) + B(3 - 3)$ , $\newline$ $8 = 4A + 0B$ , $\newline$ $A = 2$ .
Solve for A: Choose $x = -1$ to solve for A. $\newline$ Plugging $x = -1$ into the equation, we get: $\newline$ $-1 + 5 = A(-1 + 1) + B(-1 - 3)$ , $\newline$ $4 = 0A - 4B$ , $\newline$ $B = -1$ .
Write Decomposition: Write the partial fraction decomposition. $\newline$ Now that we have $A = 2$ and $B = -1$ , we can write: $\newline$ $(x+5)/(x^2-2x-3) = 2/(x-3) - 1/(x+1)$ .
Integrate Fractions: Integrate the partial fractions. $\newline$ The integral of $\frac{2}{x-3} - \frac{1}{x+1}$ is: $\newline$ $2 \cdot \ln|x-3| - \ln|x+1| + C$ , where $C$ is the constant of integration.
Combine Logarithms: Combine the logarithms. $\newline$ Using properties of logarithms, we can combine the two terms: $\newline$ $\ln|\$x-3$ ^ $2$ | - \ln|x+ $1$ | + C\).
Simplify Expression: Simplify the expression. $\newline$ The final answer is: $\newline$ $\ln\left|\frac{(x-3)^2}{(x+1)}\right| + C$ .

More problems from Evaluate definite integrals using the power rule

Question

Consider the following problem:

\newline

The water level under a bridge is changing at a rate of

r(t)=40 \sin \left(\frac{\pi t}{6}\right)

centimeters per hour (where

t

is the time in hours). At time

t=3

, the water level is

91

centimeters. By how much does the water level change during the

4^{\text {th }}

\newline

Which expression can we use to solve the problem?

\newline

1

\newline

\int_{3}^{4} r(t) d t

\newline

\int_{0}^{4} r(t) d t

\newline

\int_{4}^{5} r(t) d t

\newline

\int_{4}^{4} r(t) d t

Posted 1 month ago

Question

s(t)

gives the number of students enrolled in a school by time

t

\newline

\int_{15}^{18} s^{\prime}(t) d t=20

\newline

1

\newline

(A) The rate of change of enrollment is

20

students per year more in year

18

than it was in year

15

\newline

20

more students enrolled in year

18

15

\newline

(C) Between years

15

18

, the cumulative number of years of schooling of all of the enrolled students is

20

\newline

20

students enrolled in year

18

Posted 1 month ago

Question

A water tank is filled at a rate of

r(t)

liters per minute (where

t

is the time in minutes).

\newline

\int_{1}^{7} r^{\prime}(t) d t

\newline

1

\newline

(A) The rate at which the tank was filled at

t=7

\newline

B) The average rate of filling between

t=1

t=7

\newline

(C) The change in the rate of filling between

t=1

t=7

\newline

(D) The amount of water filled between

t=1

t=7

Posted 2 months ago

Question

The base of a solid is the region enclosed by the graphs of

y=\sin (x)

y=4-\sqrt{x}

x=2

x=7

\newline

Cross sections of the solid perpendicular to the

x

-axis are rectangles whose height is

2 x

\newline

Which one of the definite integrals gives the volume of the solid?

\newline

1

\newline

\int_{2}^{7}\left[(4-\sqrt{x})^{2}-\sin ^{2}(x)\right] d x

\newline

\int_{2}^{7}[\sin (x)+\sqrt{x}-4] \cdot 2 x d x

\newline

\int_{2}^{7}\left[\sin ^{2}(x)-(4-\sqrt{x})^{2}\right] d x

\newline

\int_{2}^{7}[4-\sqrt{x}-\sin (x)] \cdot 2 x d x

Posted 1 month ago

Question

Evaluate the integral.

\newline

\int-2 x^{3} \cos (3 x) d x

\newline

Posted 2 months ago

Question

Evaluate the integral.

\newline

\int-4 x^{2} 3^{3 x} d x

\newline

Posted 2 months ago

Question

Evaluate the integral.

\newline

\int-4 x^{2} e^{4 x} d x

\newline

Posted 2 months ago

Question

\sum_{n=1}^{\infty}\frac{3^{n}n^{2}}{n!}

Posted 2 months ago

Question

f) \lim_{n \to +\infty}\left[\frac{2}{n}+\left(\frac{3}{5}\right)^n\right]

Posted 2 months ago

Question

\int \frac{1}{\cos(x)^{2}}\,dx

Posted 2 months ago