Question

integrate (x ^ 2 + 9x + 8)/(x ^ 2 + x - 6) dx​

Answer 1

Jawab:

[tex]\displaystyle x+2\ln|x+3|+6\ln|x-2|+C[/tex]

Penjelasan dengan langkah-langkah:

Because degrees of polynomial numerator and denominator are the same, this problem solved by long division.

[tex]\begin{array}{cccc} & 1\\\cline{2-4} x^2+x-6/ & x^2 & +9x & +8\\ & x^2 & +x & -6\\\cline{2-4} & & 8x & +14\end{array}[/tex]

Definition of polynomial

f(x) = (divisor)(quotient) + remainder

In this case f(x) is dividend, so we can write

[tex]\displaystyle \rm{\frac{dividend}{divisor}=quotient+\frac{remainder}{divisor}}[/tex]

therefore

[tex]\displaystyle \frac{x^2+9x+8}{x^2+x-6}=1+\frac{8x+14}{x^2+x-6}[/tex]

Now perform partial fraction decomposition for [tex]\displaystyle \frac{8x+14}{x^2+x-6}[/tex]

[tex]\begin{aligned}\frac{8x+14}{x^2+x-6}&\:=\frac{8x+14}{(x+3)(x-2)}\\\:&=\frac{A}{x+3}+\frac{B}{x-2}\end{aligned}[/tex]

Find A and B

[tex]\begin{aligned}8x+14&\:=A(x-2)+B(x+3)\\8x+14\:&=(A+B)x+(-2A+3B)\end{aligned}[/tex]

[tex]\displaystyle A+B=8\rightarrow B=8-A[/tex]

Substitute to -2A + 3B

[tex]\begin{aligned}-2A+3B&\:=14\\-2A+3(8-A)\:&=14\\-5A+24\:&=14\\A\:&=2\end{aligned}[/tex]

then

[tex]\displaystyle B=8-2=6[/tex]

Therefore

[tex]\displaystyle \frac{8x+14}{x^2+x-6}=\frac{2}{x+3}+\frac{6}{x-2}[/tex]

Solve this problem

[tex]\begin{aligned}\int \frac{x^2+9x+8}{x^2+x-6}~dx&\:=\int \left ( 1+\frac{8x+14}{x^2+x-6} \right )dx\\\:&=\int \left ( 1+\frac{2}{x+3}+\frac{6}{x-2} \right )dx\\\:&=x+2\ln|x+3|+6\ln|x-2|+C\end{aligned}[/tex]