Integrate the following functions: 13. integral a^x dx 14. integral 1/x2 dx 15. integral x x dx 16. integral…

Question

Integrate the following functions: 13. integral a^x dx 14. integral 1/x2 dx 15. integral x x dx 16. integral (x)/(x+2) dx

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Let's tackle these integration problems.

13. $\int a^x dx =$

Apply the standard integration formula for exponential functions.
The integral of $a^x$ with respect to $x$ is $\frac{a^x}{\ln a} + C$ , where $a$ is a constant and $a > 0, a \neq 1$ .
The correct integral is $\boxed{\frac{a^x}{\ln a} + C}$ .
The correct option is (A) $\frac{a^x}{\ln a} + C$ .

14. $\int \frac{1}{x^2} dx =$
Rewrite the integrand using negative exponents.
$\int x^{-2} dx$
Apply the power rule for integration, $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ .
$\frac{x^{-2+1}}{-2+1} + C = \frac{x^{-1}}{-1} + C$
Simplify the expression.
$-\frac{1}{x} + C$
The correct integral is $\boxed{-\frac{1}{x} + C}$ .

15. $\int \sec x \tan x \, dx =$
Recall the derivative of trigonometric functions.
The derivative of $\sec x$ is $\sec x \tan x$ .
Therefore, the integral of $\sec x \tan x$ is $\sec x$ .
$\int \sec x \tan x \, dx = \sec x + C$
The correct integral is $\boxed{\sec x + C}$ .

16. $\int \frac{x}{x+2} dx =$
Rewrite the numerator to simplify the fraction.
Add and subtract 2 in the numerator.
$\int \frac{x+2-2}{x+2} dx$
Split the fraction into two terms.
$\int \left( \frac{x+2}{x+2} - \frac{2}{x+2} \right) dx$
Simplify the first term and separate the integrals.
$\int \left( 1 - \frac{2}{x+2} \right) dx = \int 1 \, dx - \int \frac{2}{x+2} dx$
Integrate each term.
$x - 2 \ln|x+2| + C$
The correct integral is $\boxed{x - 2 \ln|x+2| + C}$ .

Integrate the following functions: 13. integral a^x dx 14. integral (1)/(x2) dx 15. integral x x dx 16. integral (x)/(x+2) dx