Advertisements
Advertisements
Question
Evaluate the following.
`int ("2x" + 6)/(sqrt("x"^2 + 6"x" + 3))` dx
Advertisements
Solution
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
RELATED QUESTIONS
Find `intsqrtx/sqrt(a^3-x^3)dx`
Evaluate: `int sqrt(tanx)/(sinxcosx) dx`
Integrate the functions:
`(2x)/(1 + x^2)`
Integrate the functions:
`xsqrt(x + 2)`
Integrate the functions:
sec2(7 – 4x)
Integrate the functions:
`1/(1 + cot x)`
Write a value of\[\int e^{ax} \sin\ bx\ dx\]
Write a value of \[\int\frac{1 - \sin x}{\cos^2 x} \text{ dx }\]
The value of \[\int\frac{1}{x + x \log x} dx\] is
Evaluate the following integrals : `intsqrt(1 - cos 2x)dx`
Evaluate the following integrals : `intsqrt(1 + sin 5x).dx`
Integrate the following functions w.r.t. x : `((sin^-1 x)^(3/2))/(sqrt(1 - x^2)`
Integrate the following functions w.r.t. x : `(1 + x)/(x.sin (x + log x)`
Evaluate the following : `int (1)/(x^2 + 8x + 12).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2sin x - cosx)dx`
Evaluate `int 1/((2"x" + 3))` dx
`int cos^7 x "d"x`
`int ("d"x)/(sinx cosx + 2cos^2x)` = ______.
`int 1/(sinx.cos^2x)dx` = ______.
Evaluate `int1/(x(x - 1))dx`
Evaluate `int(1+x+x^2/(2!))dx`
Evaluate the following.
`int1/(x^2 + 4x - 5) dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
Evaluate `int(5x^2-6x+3)/(2x-3)dx`
`int (x + 1)/(x(1 + xe^x)) dx` is equal to
Which standard substitution is used for \[\sqrt{x^2+\mathrm{a}^2}\], \[\frac{1}{\sqrt{x^2+\mathrm{a}^2}}\], or \[x^2+\mathrm{a}^2\]?
Which substitution is appropriate for \[\sqrt{\frac{x}{a-x}}\], \[\sqrt{\frac{a-x}{x}}\], \[\sqrt{x(a-x)}\], or \[\frac{1}{\sqrt{x(a-x)}}\]?
What is the value of \[\int\sin^3x\cos^2x\,dx\]?
