Problem to solve
Find the indefinite integral \[\int\frac{\sqrt{x+2}-4}{5-\sqrt{x+2}} dx\]
Solution
STEP 1
Take the integral: integral(sqrt(x + 2) - 4)/(5 - sqrt(x + 2)) dx
STEP 2
For the integrand (sqrt(x + 2) - 4)/(5 - sqrt(x + 2)), substitute u = x + 2 and du = dx: = integral(sqrt(u) - 4)/(5 - sqrt(u)) du
STEP 3
For the integrand (sqrt(u) - 4)/(5 - sqrt(u)), substitute s = sqrt(u) and ds = 1/(2 sqrt(u)) du: = 2 integral(s^2 - 4 s)/(5 - s) ds
STEP 4
For the integrand (s^2 - 4 s)/(5 - s), cancel common terms in the numerator and denominator: = 2 integral(4 s - s^2)/(s - 5) ds
STEP 5
For the integrand (4 s - s^2)/(s - 5), do long division: = 2 integral-s - 5/(s - 5) - 1 ds
STEP 6
Integrate the sum term by term and factor out constants: = -2 integral s ds - 10 integral1/(s - 5) ds - 2 integral1 ds
STEP 7
The integral of s is s^2/2: = -s^2 - 10 integral1/(s - 5) ds - 2 integral1 ds
STEP 8
For the integrand 1/(s - 5), substitute p = s - 5 and dp = ds: = -s^2 - 10 integral1/p dp - 2 integral1 ds
STEP 9
The integral of 1/p is log(p): = -s^2 - 10 log(p) - 2 integral1 ds
STEP 10
The integral of 1 is s: = -10 log(p) - s^2 - 2 s + constant
STEP 11
Substitute back for p = s - 5: = -s^2 - 2 s - 10 log(s - 5) + constant
STEP 12
Substitute back for s = sqrt(u): = -u - 2 sqrt(u) - 10 log(sqrt(u) - 5) + constant
STEP 13
Substitute back for u = x + 2: = -x - 2 sqrt(x + 2) - 10 log(sqrt(x + 2) - 5) - 2 + constant
STEP 14
Factor the answer a different way: = -x - 2 (sqrt(x + 2) + 1) - 10 log(sqrt(x + 2) - 5) + constant
STEP 15
Which is equivalent for restricted x values to: Answer: | | = -x - 2 sqrt(x + 2) - 10 log(5 - sqrt(x + 2)) + 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 \(\int(-7x^5+2x^4-2)\cdot3^xdx\)
008Find the indefinite integral \(\int\arcsin^4(x) dx\)
009Find the indefinite integral \(\int\sin(\ln(x)) dx\)
010Find the indefinite integral \(\int(12x^{15}-3x^7-4x^3)\cdot e^{2x^4} dx\)
011Find the indefinite integral \(\int\frac{\sqrt{(3x^2+4)^3}}{x^4} dx\)
012Find the indefinite integral \(\int\frac{x^2}{\sqrt{2x^2-5}} dx\)
013Find the indefinite integral \(\int\frac{x^6}{\sqrt{(x^2+4)^3}} dx\)
014Find the indefinite integral \(\int x^2\sqrt{4-3x^2} dx\)
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 \(\int\frac{\sqrt{x+2}-4}{5-\sqrt{x+2}} dx\)
018Find the indefinite integral \(\int\frac{3\sqrt[3]{2x-1}-4}{\sqrt[6]{2x-1}-1} dx\)
019Find the indefinite integral \(\int\sqrt{\frac{x+3}{1-x}} dx\)
020Find the indefinite integral \(\int\frac{\sqrt[3]{x+1}-\sqrt{x+1}}{x} dx\)
021Find the indefinite integral \(\int\frac{\sin(x)-3\cos(x)}{\cos(x)-2} dx\)
022Find the indefinite integral \(\int\frac{3\sec(x)-\cot(x)}{1-\csc(x)} dx\)
023Find the indefinite integral \(\int\frac{\tan^2(x)+2}{\tan(x)+2} dx\)
024Find the indefinite integral \(\int\frac{3\sin^4(x)-2\cos^2(x)}{1+\sin^2(x)} dx\)
025Find the indefinite integral \(\int\tan^4(x)\sec^5(x) dx\)
Back
Edited by tododemate® - 2021