This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Let's tackle these integration problems. 13. a^x dx = Step 1: Apply the standard integration formula for exponential functions. The integral of a^x with respect to x is (a^x)/( a) + C, where a is a constant and a > 0, a ≠ 1. The correct integral is (a^x)/( a) + C. The correct option is (A) (a^x)/( a) + C. 14. (1)/(x^2) dx = Step 1: Rewrite the integrand using negative exponents. x^-2 dx Step 2: Apply the power rule for integration, x^n dx = x^n+1n+1 + C. x^-2+1-2+1 + C = x^-1-1 + C Step 3: Simplify the expression. -(1)/(x) + C The correct integral is -(1)/(x) + C. 15. x x \, dx = Step 1: Recall the derivative of trigonometric functions. The derivative of x is x x. Step 2: Therefore, the integral of x x is x. x x \, dx = x + C The correct integral is x + C. 16. (x)/(x+2) dx = Step 1: Rewrite the numerator to simplify the fraction. Add and subtract 2 in the numerator. (x+2-2)/(x+2) dx Step 2: Split the fraction into two terms. ( (x+2)/(x+2) - (2)/(x+2) ) dx Step 3: Simplify the first term and separate the integrals. ( 1 - (2)/(x+2) ) dx = 1 \, dx - (2)/(x+2) dx Step 4: Integrate each term. x - 2 |x+2| + C The correct integral is x - 2 |x+2| + C. Last free one today — make it count tomorrow, or type /upgrade for unlimited.