Advertisements
Advertisements
प्रश्न
Integrate the following with respect to the respective variable:
`x^7/(x + 1)`
Advertisements
उत्तर
Let I = `int x^7/(x + 1)*dx`
= `int ((x^7 + 1) - 1)/(x + 1)*dx`
= `int ((x + 1)(x^6 - x^5 + x^4 - x^3 + x^2 - x + 1) - 1)/(x + 1)*dx`
= `int [x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 - (1)/(x + 1)]*dx`
= `int x^6 dx - intx^5 dx + intx^4 dx - intx^3 dx + intx^2 dx - intx dx + int1 dx - int (1)/(x + 1) dx`
= `x^7/(7) - x^6/(6) + x^5/(5) - x^4/(4) + x^3/(3) - x^2/(2) + x - log|x + 1| + c`.
APPEARS IN
संबंधित प्रश्न
Evaluate :
`int(sqrt(cotx)+sqrt(tanx))dx`
Integrate the functions:
`e^(2x+3)`
Integrate the functions:
`(sin^(-1) x)/(sqrt(1-x^2))`
Integrate the functions:
`1/(cos^2 x(1-tan x)^2`
Write a value of
Write a value of\[\int\frac{\sin x}{\cos^3 x} \text{ dx }\]
Write a value of\[\int\frac{1}{x \left( \log x \right)^n} \text { dx }\].
Evaluate the following integrals : `intsqrt(1 - cos 2x)dx`
Integrate the following function w.r.t. x:
x9.sec2(x10)
Integrate the following functions w.r.t. x : `(x^n - 1)/sqrt(1 + 4x^n)`
Integrate the following functions w.r.t. x:
`x^5sqrt(a^2 + x^2)`
Integrate the following functions w.r.t. x : `(7 + 4 + 5x^2)/(2x + 3)^(3/2)`
Evaluate the following:
`int (1)/(25 - 9x^2)*dx`
Evaluate the following : `int (1)/sqrt(2x^2 - 5).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2 sin2x + 4cos 2x).dx`
Choose the correct options from the given alternatives :
`int sqrt(cotx)/(sinx*cosx)*dx` =
Integrate the following with respect to the respective variable : `(x - 2)^2sqrt(x)`
If f'(x) = x2 + 5 and f(0) = −1, then find the value of f(x).
Evaluate the following.
`int (1 + "x")/("x" + "e"^"-x")` dx
Evaluate the following.
`int 1/("x" log "x")`dx
Evaluate the following.
`int ("2x" + 6)/(sqrt("x"^2 + 6"x" + 3))` dx
Evaluate the following.
`int 1/(7 + 6"x" - "x"^2)` dx
`int sqrt(1 + "x"^2) "dx"` =
Fill in the Blank.
`int (5("x"^6 + 1))/("x"^2 + 1)` dx = x4 + ______ x3 + 5x + c
Fill in the Blank.
`int 1/"x"^3 [log "x"^"x"]^2 "dx" = "P" (log "x")^3` + c, then P = _______
Evaluate:
`int (5x^2 - 6x + 3)/(2x − 3)` dx
`int (log x)/(log ex)^2` dx = _________
`int x/(x + 2) "d"x`
`int(log(logx))/x "d"x`
`int "e"^(sin^-1 x) ((x + sqrt(1 - x^2))/(sqrt1 - x^2)) "dx" = ?`
`int ("e"^x(x + 1))/(sin^2(x"e"^x)) "d"x` = ______.
`int (f^'(x))/(f(x))dx` = ______ + c.
`int 1/(sinx.cos^2x)dx` = ______.
Evaluate `int(1 + x + x^2/(2!) )dx`
Evaluate the following.
`int x^3/(sqrt(1+x^4))dx`
Evaluate the following.
`int(20 - 12"e"^"x")/(3"e"^"x" - 4) "dx"`
Evaluate:
`int 1/(1 + cosα . cosx)dx`
Evaluate:
`int(sqrt(tanx) + sqrt(cotx))dx`
`int "cosec"^4x dx` = ______.
`int (cos4x)/(sin2x + cos2x)dx` = ______.
Evaluate the following.
`intxsqrt(1+x^2)dx`
Evaluate `int1/(x(x-1))dx`
Evaluate `int1/(x(x-1))dx`
Evaluate `int (5x^2 - 6x + 3)/(2x - 3) dx`
In \[\int\sin^3x\cos^2x\,dx\], which rewriting prepares the integrand for the substitution \[t=\cos x\]?
For \[t=\cos x\], what is \[dt\]?
What is \[\int\cot t\,dt\] in the evaluation of \[\int\frac{\sin x}{\sin(x+a)}\,dx\]?
Integration by substitution is the reverse process of which rule?
