Advertisements
Advertisements
प्रश्न
Evaluate :
`∫(x+2)/sqrt(x^2+5x+6)dx`
Advertisements
उत्तर
`I=∫(x+2)/sqrt(x^2+5x+6)dx `
Multiplying and dividing by 2, we get
`I=1/2∫(2x+4)/sqrt(x^2+5x+6)dx `
Adding and subtracting 1 to the numerator, we get:
`I=1/2∫(2x+4+1-1)/sqrt(x^2+5x+6)dx`
` I=1/2∫(2x+5)/sqrt(x2+5x+6)dx -1/2∫1/sqrt(x^2+5x+6)dx`
`"Let" I_1=1/2∫(2x+5)/sqrt(x^2+5x+6)dx `
Put x2+5x+6=t
Differentiating with respect to x, we get:
(2x+5)dx=dt
`I_1=intdt/sqrtt`
`I_1=2sqrtt+c`
`I_1=2sqrt(x^2+5x+6)+c`
`1/2 int "dt"/sqrt t =∫1/sqrt(x^2+5x+(5/2)^2-(5/2)^2+6)dx`
`1/2 int "dt"/sqrt t - 1/2 int "dx"/sqrt(x^2+5x+6 + (5/2)^2 - 25/4)dx`
`1/2 "t"^(1/2)/(1/2) int 1/sqrt((x+5/2)^2-(1/2)^2)dx`
`= 1/2 xx 2 xx "t"^(1/2) - 1/2 |"log" x + 5/2 + sqrt (x^2 + 5x + 6)| + "C"`
`= sqrt (x^2 + 5x + 6) - 1/2 "log" |sqrt (x^2 + 5x + 6)| + "C"`
APPEARS IN
संबंधित प्रश्न
Evaluate : `int (sinx)/sqrt(36-cos^2x)dx`
Integrate the functions:
sin x ⋅ sin (cos x)
Integrate the functions:
`xsqrt(1+ 2x^2)`
Integrate the functions:
`(e^(2x) - e^(-2x))/(e^(2x) + e^(-2x))`
Integrate the functions:
tan2(2x – 3)
Evaluate: `int_0^3 f(x)dx` where f(x) = `{(cos 2x, 0<= x <= pi/2),(3, pi/2 <= x <= 3) :}`
Write a value of
Write a value of\[\int\frac{1}{x \left( \log x \right)^n} \text { dx }\].
Evaluate the following integrals : `intsqrt(1 + sin 5x).dx`
Integrate the following functions w.r.t. x : `sin(x - a)/cos(x + b)`
Integrate the following functions w.r.t. x : `3^(cos^2x) sin 2x`
Evaluate the following : `int (1)/(7 + 2x^2).dx`
Evaluate the following : `int (1)/sqrt(2x^2 - 5).dx`
Evaluate the following : `int (1)/(1 + x - x^2).dx`
Evaluate the following integral:
`int (3cosx)/(4sin^2x + 4sinx - 1).dx`
Evaluate the following.
`int 1/(sqrt(3"x"^2 - 5))` dx
Evaluate the following.
`int 1/(sqrt("x"^2 -8"x" - 20))` dx
Choose the correct alternative from the following.
The value of `int "dx"/sqrt"1 - x"` is
`int sqrt(x^2 + 2x + 5)` dx = ______________
`int cos sqrtx` dx = _____________
`int cot^2x "d"x`
`int "dx"/((sin x + cos x)(2 cos x + sin x))` = ?
`int "cosec"^4x dx` = ______.
Evaluate the following.
`intxsqrt(1+x^2)dx`
Evaluate the following
`int x^3 e^(x^2) ` dx
Evaluate the following.
`int1/(x^2+4x-5)dx`
Evaluate `int 1/(x(x-1))dx`
`int (x + 1)/(x(1 + xe^x)) dx` is equal to
