evaluate the following integral.\n\n∫x + 2x² + 4x + 49dx\n\n∫x + 2x² + 4x + 49dx = □\n\n(use parentheses to…

evaluate the following integral.\n\n∫x + 2x² + 4x + 49dx\n\n∫x + 2x² + 4x + 49dx = □\n\n(use parentheses to clearly denote the argument of each function.)

evaluate the following integral.\n\n∫x + 2x² + 4x + 49dx\n\n∫x + 2x² + 4x + 49dx = □\n\n(use parentheses to clearly denote the argument of each function.)

Answer

Explanation:

Step1: Split the integrand

$$\int\frac{x + 2}{x^{2}+4x + 49}dx=\int\frac{x + 2}{(x + 2)^{2}+45}dx$$ Let (u=(x + 2)^{2}+45), then (du = 2(x + 2)dx), and ((x + 2)dx=\frac{1}{2}du)

Step2: Integrate

$$\int\frac{x + 2}{(x + 2)^{2}+45}dx=\frac{1}{2}\int\frac{du}{u}$$ Using the formula (\int\frac{1}{u}du=\ln|u|+C) $$\frac{1}{2}\int\frac{du}{u}=\frac{1}{2}\ln|u|+C=\frac{1}{2}\ln\left(x^{2}+4x + 49\right)+C$$

Answer:

(\frac{1}{2}\ln\left(x^{2}+4x + 49\right)+C)