evaluate the following integral\n int \frac { 4 x ^ { 2 } + x + 1 7 } { ( x + 1 ) ( x ^ { 2 } + 4 ) } d x…

evaluate the following integral\n int \frac { 4 x ^ { 2 } + x + 1 7 } { ( x + 1 ) ( x ^ { 2 } + 4 ) } d x \ncan partial fraction decomposition be used to evaluate the given integral? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. yes, partial fraction decomposition can be used. the given integral can be rewritten as int left \frac { 1 } { x + 1 } + \frac { 4 } { x ^ { 2 } + 4 } \right d x , which is more readily evaluated.\nb. no, partial fraction decomposition cannot be used.\nevaluate the indefinite integral\n int \frac { 4 x ^ { 2 } + x + 1 7 } { ( x + 1 ) ( x ^ { 2 } + 4 ) } d x = \n(use parentheses to clearly denote the argument of each function.)

evaluate the following integral\n int \frac { 4 x ^ { 2 } + x + 1 7 } { ( x + 1 ) ( x ^ { 2 } + 4 ) } d x \ncan partial fraction decomposition be used to evaluate the given integral? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. yes, partial fraction decomposition can be used. the given integral can be rewritten as int left \frac { 1 } { x + 1 } + \frac { 4 } { x ^ { 2 } + 4 } \right d x , which is more readily evaluated.\nb. no, partial fraction decomposition cannot be used.\nevaluate the indefinite integral\n int \frac { 4 x ^ { 2 } + x + 1 7 } { ( x + 1 ) ( x ^ { 2 } + 4 ) } d x = \n(use parentheses to clearly denote the argument of each function.)

Answer

Explanation:

Step1: Integrate term - by - term

$$\int\left(\frac{1}{x + 1}+\frac{4}{x^{2}+4}\right)dx=\int\frac{1}{x + 1}dx+\int\frac{4}{x^{2}+4}dx$$

Step2: Integrate $\int\frac{1}{x + 1}dx$

Use the formula $\int\frac{1}{u}du=\ln|u|+C$. Let $u=x + 1$, then $du=dx$. So, $\int\frac{1}{x + 1}dx=\ln|x + 1|+C_1$.

Step3: Integrate $\int\frac{4}{x^{2}+4}dx$

Use the formula $\int\frac{1}{a^{2}+x^{2}}dx=\frac{1}{a}\arctan(\frac{x}{a})+C$. Here $a = 2$, so $\int\frac{4}{x^{2}+4}dx=4\times\frac{1}{2}\arctan(\frac{x}{2})+C_2 = 2\arctan(\frac{x}{2})+C_2$.

Step4: Combine the results

Combine the results of the two integrals: $\int\left(\frac{1}{x + 1}+\frac{4}{x^{2}+4}\right)dx=\ln|x + 1|+2\arctan(\frac{x}{2})+C$, where $C = C_1+C_2$.

Answer:

$\ln|x + 1|+2\arctan(\frac{x}{2})+C$