Problem to solve
Find the indefinite integral \[\int\frac{x^2+2}{x^3+5x^2+2x-8} dx\]
Solution
STEP 1
Take the integral: integral(x^2 + 2)/(x^3 + 5 x^2 + 2 x - 8) dx
STEP 2
For the integrand (x^2 + 2)/(x^3 + 5 x^2 + 2 x - 8), use partial fractions: = integral-1/(x + 2) + 9/(5 (x + 4)) + 1/(5 (x - 1)) dx
STEP 3
Integrate the sum term by term and factor out constants: = 9/5 integral1/(x + 4) dx - integral1/(x + 2) dx + 1/5 integral1/(x - 1) dx
STEP 4
For the integrand 1/(x + 4), substitute u = x + 4 and du = dx: = 9/5 integral1/u du - integral1/(x + 2) dx + 1/5 integral1/(x - 1) dx
STEP 5
The integral of 1/u is log(u): = (9 log(u))/5 - integral1/(x + 2) dx + 1/5 integral1/(x - 1) dx
STEP 6
For the integrand 1/(x + 2), substitute s = x + 2 and ds = dx: = (9 log(u))/5 - integral1/s ds + 1/5 integral1/(x - 1) dx
STEP 7
The integral of 1/s is log(s): = -log(s) + (9 log(u))/5 + 1/5 integral1/(x - 1) dx
STEP 8
For the integrand 1/(x - 1), substitute p = x - 1 and dp = dx: = -log(s) + (9 log(u))/5 + 1/5 integral1/p dp
STEP 9
The integral of 1/p is log(p): = log(p)/5 - log(s) + (9 log(u))/5 + constant
STEP 10
Substitute back for p = x - 1: = -log(s) + (9 log(u))/5 + 1/5 log(x - 1) + constant
STEP 11
Substitute back for s = x + 2: = (9 log(u))/5 + 1/5 log(x - 1) - log(x + 2) + constant
STEP 12
Substitute back for u = x + 4: = 1/5 log(x - 1) - log(x + 2) + 9/5 log(x + 4) + constant
STEP 13
Factor the answer a different way: = 1/5 (log(x - 1) - 5 log(x + 2) + 9 log(x + 4)) + constant
STEP 14
Which is equivalent for restricted x values to: Answer: | | = 1/5 (log(1 - x) - 5 log(x + 2) + 9 log(x + 4)) + constant
More problems
Number
001Find the indefinite integral \(\int\frac{x^2+2}{x^3+5x^2+2x-8} dx\)
002Find the indefinite integral \(\int\frac{4x^4-2}{x^3+4x^2+x-6} dx\)
003Find the indefinite integral \(\int\frac{4x^4-22x^3+11x^2-99x-34}{x^3-6x^2+4x-24} dx\)
004Find the indefinite integral \(\int\frac{3x^4+4x^2+5x}{(x-1)(x^2+1)^2} dx\)
005Find the indefinite integral \(\int\frac{x^6+6x^4+8x^2-4}{(x^2+4)(x^2+3)} dx\)
006Find the indefinite integral \(\int(-4x^4+6x^2+8x-1)\cos(x) dx\)
007Find the indefinite integral \(\int(-7x^5+2x^4-2)\cdot3^xdx\)
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015Find the indefinite integral \(\int x^4\cdot\sqrt{x^2+4x+5} dx\)
016Find the indefinite integral \(\int\frac{\sqrt{x+2}}{x+1} dx\)
017Find the indefinite integral \(\int\frac{\sqrt{x+2}-4}{5-\sqrt{x+2}} dx\)
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019Find the indefinite integral \(\int\sqrt{\frac{x+3}{1-x}} dx\)
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023Find the indefinite integral \(\int\frac{\tan^2(x)+2}{\tan(x)+2} dx\)
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025Find the indefinite integral \(\int\tan^4(x)\sec^5(x) dx\)
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Edited by tododemate® - 2021