Advertisements
Advertisements
Question
Find: `int_ (3"x"+ 5)sqrt(5 + 4"x"-2"x"^2)d"x"`.
Advertisements
Solution
`( 3x + 5)sqrt(5 + 4x - 2x^2) dx`
Let 3x + 5 = A(4 - 4x) + B
⇒ A = `-3/4, B = 8`
I = `3/4(4 - 4x)sqrt(5 + 4x + 2x^2) dx + 8 sqrt(5 + 4x - 2x^2)dx`
= `-3/4 I_1 + 8l_2 ("let")`
For I1, put 5 + 4x = - 2x2 = t
⇒ (4 - 4x) dx = dt
`-3/4 I_1 = - 3/4 sqrtt dt = - 3/4 xx 2/3 t^(3/2)`
= `-1/2 (5 + 4x - 2x^2 )^(3/2)`
`8I_2 = 8sqrt2 sqrt(7/2 - ( x - 1)^2) dx`
I = `-1/2 (5 + 4x - 2x^2 )^(3/2) + 4sqrt2(x - 1) sqrt(5/2 + 2x - x^2) + 14sqrt2 sin^-1 (sqrt2(x - 1))/sqrt7 + C`
APPEARS IN
RELATED QUESTIONS
Evaluate `∫_0^(3/2)|x cosπx|dx`
If `int_0^a1/(4+x^2)dx=pi/8` , find the value of a.
Evaluate the integral by using substitution.
`int_0^1 x/(x^2 +1)`dx
Evaluate of the following integral:
Evaluate of the following integral:
Evaluate:
Evaluate the following definite integral:
Evaluate the following integral:
Evaluate the following integral:
Evaluate the following integral:
Evaluate the following integral:
Evaluate each of the following integral:
Evaluate the following integral:
Evaluate the following integral:
Evaluate : \[\int\limits_{- 2}^1 \left| x^3 - x \right|dx\] .
Find : \[\int\frac{x \sin^{- 1} x}{\sqrt{1 - x^2}}dx\] .
Evaluate: `int_ e^x ((2+sin2x))/cos^2 x dx`
If `I_n = int_0^(pi/4) tan^n theta "d"theta " then " I_8 + I_6` equals ______.
`int_0^1 x(1 - x)^5 "dx" =` ______.
`int_0^(pi4) sec^4x "d"x` = ______.
Evaluate the following:
`int "dt"/sqrt(3"t" - 2"t"^2)`
Find: `int (dx)/sqrt(3 - 2x - x^2)`
Evaluate: `int_0^(π/2) sin 2x tan^-1 (sin x) dx`.
Evaluate: `int x/(x^2 + 1)"d"x`
In Method: Resubstitution, what should be done first?
After finding the antiderivative in Method: Resubstitution, what is the next step?
Which limits are used to evaluate the resulting expression in Method: Resubstitution?
If \[t=\tan^{-1}x\], what is \[dt\]?
After using \[t=\tan^{-1}x\] in \[\int_{0}^{1}\frac{\tan^{-1}x}{1+x^2}\,dx\], which integral results?
