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Problem to solve

Find the indefinite integral \[\int\sqrt{\frac{x+3}{1-x}} dx\]

Solution

STEP 1

Take the integral:
integral sqrt((x + 3)/(1 - x)) dx

STEP 2

For the integrand sqrt((x + 3)/(1 - x)), substitute u = (x + 3)/(1 - x) and du = (1/(1 - x) - (-x - 3)/(1 - x)^2) dx:
= 4 integral sqrt(u)/(u + 1)^2 du

STEP 3

For the integrand sqrt(u)/(u + 1)^2, substitute s = sqrt(u) and ds = 1/(2 sqrt(u)) du:
= 8 integral s^2/(s^2 + 1)^2 ds

STEP 4

$For the integrand s^2/(s^2 + 1)^2, use partial fractions: = 8 integral1/(s^2 + 1) - 1/(s^2 + 1)^2 ds$

STEP 5

Integrate the sum term by term and factor out constants:
= 8 integral1/(s^2 + 1) ds - 8 integral1/(s^2 + 1)^2 ds

STEP 6

The integral of 1/(s^2 + 1) is tan^(-1)(s):
= 8 tan^(-1)(s) - 8 integral1/(s^2 + 1)^2 ds

STEP 7

For the integrand 1/(s^2 + 1)^2, substitute s = tan(p) and ds = sec^2(p) dp. Then (s^2 + 1)^2 = (tan^2(p) + 1)^2 = sec^4(p) and p = tan^(-1)(s):
= 8 tan^(-1)(s) - 8 integral cos^2(p) dp

STEP 8

Write cos^2(p) as 1/2 cos(2 p) + 1/2:
= 8 tan^(-1)(s) - 8 integral(1/2 cos(2 p) + 1/2) dp

STEP 9

Integrate the sum term by term and factor out constants:
= 8 tan^(-1)(s) - 4 integral cos(2 p) dp - 4 integral1 dp

STEP 10

For the integrand cos(2 p), substitute w = 2 p and dw = 2 dp:
= 8 tan^(-1)(s) - 2 integral cos(w) dw - 4 integral1 dp

STEP 11

The integral of cos(w) is sin(w):
= 8 tan^(-1)(s) - 2 sin(w) - 4 integral1 dp

STEP 12

The integral of 1 is p:
= -4 p + 8 tan^(-1)(s) - 2 sin(w) + constant

STEP 13

Substitute back for w = 2 p:
= -4 p - 2 sin(2 p) + 8 tan^(-1)(s) + constant

STEP 14

Substitute back for p = tan^(-1)(s):
= 4 tan^(-1)(s) - (4 s)/(s^2 + 1) + constant

STEP 15

Substitute back for s = sqrt(u):
= 4 tan^(-1)(sqrt(u)) - (4 sqrt(u))/(u + 1) + constant

STEP 16

Substitute back for u = (x + 3)/(1 - x):
= sqrt((x + 3)/(1 - x)) (x - 1) + 4 tan^(-1)(sqrt((x + 3)/(1 - x))) + constant

STEP 17

Which is equivalent for restricted x values to:
Answer: |
| = (sqrt((x + 3)/(1 - x)) ((x - 1) sqrt(x + 3) - 4 sqrt(1 - x) sin^(-1)(sqrt(1 - x)/2)))/sqrt(x + 3) + constant

More problems

Number
001	Find the indefinite integral $\int\frac{x^2+2}{x^3+5x^2+2x-8} dx$
002	Find the indefinite integral $\int\frac{4x^4-2}{x^3+4x^2+x-6} dx$
003	Find the indefinite integral $\int\frac{4x^4-22x^3+11x^2-99x-34}{x^3-6x^2+4x-24} dx$
004	Find the indefinite integral $\int\frac{3x^4+4x^2+5x}{(x-1)(x^2+1)^2} dx$
005	Find the indefinite integral $\int\frac{x^6+6x^4+8x^2-4}{(x^2+4)(x^2+3)} dx$
006	Find the indefinite integral $\int(-4x^4+6x^2+8x-1)\cos(x) dx$
007	Find the indefinite integral $\int(-7x^5+2x^4-2)\cdot3^xdx$
008	Find the indefinite integral $\int\arcsin^4(x) dx$
009	Find the indefinite integral $\int\sin(\ln(x)) dx$
010	Find the indefinite integral $\int(12x^{15}-3x^7-4x^3)\cdot e^{2x^4} dx$
011	Find the indefinite integral $\int\frac{\sqrt{(3x^2+4)^3}}{x^4} dx$
012	Find the indefinite integral $\int\frac{x^2}{\sqrt{2x^2-5}} dx$
013	Find the indefinite integral $\int\frac{x^6}{\sqrt{(x^2+4)^3}} dx$
014	Find the indefinite integral $\int x^2\sqrt{4-3x^2} dx$
015	Find the indefinite integral $\int x^4\cdot\sqrt{x^2+4x+5} dx$
016	Find the indefinite integral $\int\frac{\sqrt{x+2}}{x+1} dx$
017	Find the indefinite integral $\int\frac{\sqrt{x+2}-4}{5-\sqrt{x+2}} dx$
018	Find the indefinite integral $\int\frac{3\sqrt[3]{2x-1}-4}{\sqrt[6]{2x-1}-1} dx$
019	Find the indefinite integral $\int\sqrt{\frac{x+3}{1-x}} dx$
020	Find the indefinite integral $\int\frac{\sqrt[3]{x+1}-\sqrt{x+1}}{x} dx$
021	Find the indefinite integral $\int\frac{\sin(x)-3\cos(x)}{\cos(x)-2} dx$
022	Find the indefinite integral $\int\frac{3\sec(x)-\cot(x)}{1-\csc(x)} dx$
023	Find the indefinite integral $\int\frac{\tan^2(x)+2}{\tan(x)+2} dx$
024	Find the indefinite integral $\int\frac{3\sin^4(x)-2\cos^2(x)}{1+\sin^2(x)} dx$
025	Find the indefinite integral $\int\tan^4(x)\sec^5(x) dx$

Back

Number
001	Find the indefinite integral \(\int\frac{x^2+2}{x^3+5x^2+2x-8} dx\)
002	Find the indefinite integral \(\int\frac{4x^4-2}{x^3+4x^2+x-6} dx\)
003	Find the indefinite integral \(\int\frac{4x^4-22x^3+11x^2-99x-34}{x^3-6x^2+4x-24} dx\)
004	Find the indefinite integral \(\int\frac{3x^4+4x^2+5x}{(x-1)(x^2+1)^2} dx\)
005	Find the indefinite integral \(\int\frac{x^6+6x^4+8x^2-4}{(x^2+4)(x^2+3)} dx\)
006	Find the indefinite integral \(\int(-4x^4+6x^2+8x-1)\cos(x) dx\)
007	Find the indefinite integral \(\int(-7x^5+2x^4-2)\cdot3^xdx\)
008	Find the indefinite integral \(\int\arcsin^4(x) dx\)
009	Find the indefinite integral \(\int\sin(\ln(x)) dx\)
010	Find the indefinite integral \(\int(12x^{15}-3x^7-4x^3)\cdot e^{2x^4} dx\)
011	Find the indefinite integral \(\int\frac{\sqrt{(3x^2+4)^3}}{x^4} dx\)
012	Find the indefinite integral \(\int\frac{x^2}{\sqrt{2x^2-5}} dx\)
013	Find the indefinite integral \(\int\frac{x^6}{\sqrt{(x^2+4)^3}} dx\)
014	Find the indefinite integral \(\int x^2\sqrt{4-3x^2} dx\)
015	Find the indefinite integral \(\int x^4\cdot\sqrt{x^2+4x+5} dx\)
016	Find the indefinite integral \(\int\frac{\sqrt{x+2}}{x+1} dx\)
017	Find the indefinite integral \(\int\frac{\sqrt{x+2}-4}{5-\sqrt{x+2}} dx\)
018	Find the indefinite integral \(\int\frac{3\sqrt[3]{2x-1}-4}{\sqrt[6]{2x-1}-1} dx\)
019	Find the indefinite integral \(\int\sqrt{\frac{x+3}{1-x}} dx\)
020	Find the indefinite integral \(\int\frac{\sqrt[3]{x+1}-\sqrt{x+1}}{x} dx\)
021	Find the indefinite integral \(\int\frac{\sin(x)-3\cos(x)}{\cos(x)-2} dx\)
022	Find the indefinite integral \(\int\frac{3\sec(x)-\cot(x)}{1-\csc(x)} dx\)
023	Find the indefinite integral \(\int\frac{\tan^2(x)+2}{\tan(x)+2} dx\)
024	Find the indefinite integral \(\int\frac{3\sin^4(x)-2\cos^2(x)}{1+\sin^2(x)} dx\)
025	Find the indefinite integral \(\int\tan^4(x)\sec^5(x) dx\)