Solve the following two math problems: 1. If f(x) = (1)/(x+1), find f(x−1). 2. Integrate the function y = 3x2…

Question

Solve the following two math problems: 1. If f(x) = (1)/(x+1), find f(x−1). 2. Integrate the function y = 3x2 + 2x + 5 with respect to x.

ScanSolve AI Tutor · Accepted Answer

Here are the solutions to the problems.

:
If $f(x) = \frac{1}{x+1}$ , find $f(x-1)$ .

Substitute $(x-1)$ for $x$ in the function $f(x)$ .
$f(x-1) = \frac{1}{(x-1)+1}$
Simplify the expression.
$f(x-1) = \frac{1}{x}$
The function $f(x-1)$ is:
$\boxed{\frac{1}{x}}$
:
Integrate the function $y = 3x^2 + 2x + 5$ with respect to $x$ .
Set up the integral.
$\int (3x^2 + 2x + 5)\,dx$
Apply the power rule for integration $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ to each term.
$\int 3x^2\,dx + \int 2x\,dx + \int 5\,dx$ $= 3\left(\frac{x^{2+1}}{2+1}\right) + 2\left(\frac{x^{1+1}}{1+1}\right) + 5x + C$ $= 3\left(\frac{x^3}{3}\right) + 2\left(\frac{x^2}{2}\right) + 5x + C$
Simplify the expression.
$= x^3 + x^2 + 5x + C$
The integral of the function is:
$\boxed{x^3 + x^2 + 5x + C}$

Solve the following two math problems: 1. If f(x) = (1)/(x+1), find f(x-1). 2. Integrate the function y = 3x2 + 2x + 5 with respect to x.