Evaluate the following integrals: 16. integral x(x + 1/x) dx 17. integral (x2 + 2x + 1)/(x+1) dx

Question

ScanSolve AI Tutor · Accepted Answer

Here are the solutions for the integration problems:

16.
To evaluate the integral $\int x\left(x + \frac{1}{x}\right)\,dx$ :

Distribute $x$ into the parenthesis.
$x\left(x + \frac{1}{x}\right) = x \cdot x + x \cdot \frac{1}{x} = x^2 + 1$
Integrate the simplified expression term by term.
$\int (x^2 + 1)\,dx = \int x^2\,dx + \int 1\,dx$
Apply the power rule for integration $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ .
$\int x^2\,dx = \frac{x^{2+1}}{2+1} + C_1 = \frac{x^3}{3} + C_1$ $\int 1\,dx = x + C_2$
Combine the results and add the constant of integration $C$ .
$\int x\left(x + \frac{1}{x}\right)\,dx = \frac{x^3}{3} + x + C$
The integral is $\boxed{\frac{x^3}{3} + x + C}$ .

17.
To evaluate the integral $\int \frac{x^2 + 2x + 1}{x+1}\,dx$ :
Factor the numerator.
The numerator $x^2 + 2x + 1$ is a perfect square trinomial, which can be factored as $(x+1)^2$ .
$\frac{x^2 + 2x + 1}{x+1} = \frac{(x+1)^2}{x+1}$
Simplify the expression by canceling out one $(x+1)$ term, assuming $x \neq -1$ .
$\frac{(x+1)^2}{x+1} = x+1$
Integrate the simplified expression term by term.
$\int (x+1)\,dx = \int x\,dx + \int 1\,dx$
Apply the power rule for integration.
$\int x\,dx = \frac{x^{1+1}}{1+1} + C_1 = \frac{x^2}{2} + C_1$ $\int 1\,dx = x + C_2$
Combine the results and add the constant of integration $C$ .
$\int \frac{x^2 + 2x + 1}{x+1}\,dx = \frac{x^2}{2} + x + C$
The integral is $\boxed{\frac{x^2}{2} + x + C}$ .

Evaluate the following integrals: 16. integral x(x + (1)/(x))\,dx 17. integral (x2 + 2x + 1)/(x+1)\,dx