Advertisements
Advertisements
प्रश्न
Evaluate the following.
`int ("2x" + 6)/(sqrt("x"^2 + 6"x" + 3))` dx
Advertisements
उत्तर
Let I = `int ("2x" + 6)/(sqrt("x"^2 + 6"x" + 3))` dx
Put x2 + 6x + 3 = t
∴ (2x + 6) dx = dt
∴ I = `int "dt"/sqrt"t"`
`= int "t"^((-1)/2)`dt
`= "t"^(1/2)/(1/2)` + c
`= 2 sqrt"t"` + c
∴ I = `2 sqrt("x"^2 + "6x" + 3)` + c
Alternate Method:
Let I = `int ("2x" + 6)/(sqrt("x"^2 + 6"x" + 3))` dx
`"d"/"dx" ("x"^2 + "6x" + 3)` = 2x + 6
∴ I = `int ("d"/"dx" ("x"^2 + "6x" + 3))/(sqrt("x"^2 + 6"x" + 3))` dx
∴ I = `2 sqrt("x"^2 + "6x" + 3)` + c ....`[because int ("f" '("x"))/sqrt("f"("x")) "dx" = 2sqrt("f"("x")) + "c"]`
APPEARS IN
संबंधित प्रश्न
Prove that `int_a^bf(x)dx=f(a+b-x)dx.` Hence evaluate : `int_a^bf(x)/(f(x)+f(a-b-x))dx`
Integrate the functions:
`(log x)^2/x`
Integrate the functions:
`1/(x-sqrtx)`
Integrate the functions:
`sqrt(sin 2x) cos 2x`
Evaluate: `int (2y^2)/(y^2 + 4)dx`
Write a value of\[\int\frac{1}{1 + 2 e^x} \text{ dx }\].
Write a value of
Write a value of\[\int\sqrt{4 - x^2} \text{ dx }\]
\[\int\frac{\sin x + 2 \cos x}{2 \sin x + \cos x} \text{ dx }\]
Evaluate the following integrals : `int (sin2x)/(cosx)dx`
Evaluate the following integral:
`int(4x + 3)/(2x + 1).dx`
Integrate the following functions w.r.t. x : `((sin^-1 x)^(3/2))/(sqrt(1 - x^2)`
Evaluate the following : `int (1)/(4x^2 - 3).dx`
Evaluate the following : `int (1)/(5 - 4x - 3x^2).dx`
Integrate the following functions w.r.t. x : `int (1)/(4 - 5cosx).dx`
Integrate the following functions w.r.t. x : `int (1)/(2 + cosx - sinx).dx`
Integrate the following functions w.r.t. x : `int (1)/(2sin 2x - 3)dx`
Evaluate the following integrals : `int (2x + 3)/(2x^2 + 3x - 1).dx`
Evaluate the following.
`int (1 + "x")/("x" + "e"^"-x")` dx
Evaluate the following.
`int 1/("x" log "x")`dx
`int ("e"^(3x))/("e"^(3x) + 1) "d"x`
`int cot^2x "d"x`
Evaluate `int(3x^2 - 5)^2 "d"x`
`int x^3"e"^(x^2) "d"x`
`int 1/(sinx.cos^2x)dx` = ______.
Evaluate `int (1+x+x^2/(2!))dx`
Evaluate the following:
`int (1) / (x^2 + 4x - 5) dx`
Evaluate the following.
`int1/(x^2 + 4x-5)dx`
Evaluate the following.
`intx^3/sqrt(1 + x^4) dx`
