Advertisements
Advertisements
Question
Integrate the function:
`(sin^8 x - cos^8 x)/(1-2sin^2 x cos^2 x)`
Advertisements
Solution
Let `I = int (sin^8 x - cos^8 x)/ (1 - 2 sin^2x cos^2 x) dx`
We have, (sin8x - cos8x)
= (sin4x + cos4x) (sin4x - cos4x)
= [(sin2x + cos2x)2 - 2 sin2x cos2 x] (sin2 x + cos2 x) (sin2 x - cos2 x)
= (1 - 2 sin2x cos2x) (1) (-cos2x)
∴ `I = int ((1 - 2 sin^2 x cos^2 x) (- cos 2x))/(1 - 2 sin^2 x cos x) dx`
`= - int cos 2x dx`
`= -1/2 sin2x + C`
APPEARS IN
RELATED QUESTIONS
Write the antiderivative of `(3sqrtx+1/sqrtx).`
Find :`int(x^2+x+1)/((x^2+1)(x+2))dx`
Find an anti derivative (or integral) of the following function by the method of inspection.
e2x
Find the following integrals:
`int (ax^2 + bx + c) dx`
Find the following integrals:
`int(2x^2 + e^x)dx`
Find the following integrals:
`int (x^3 + 3x + 4)/sqrtx dx`
Integrate the function:
`1/(x^(1/2) + x^(1/3)) ["Hint:" 1/(x^(1/2) + x^(1/3)) = 1/(x^(1/3)(1+x^(1/6))), "put x" = t^6]`
Integrate the function:
`(e^(5log x) - e^(4log x))/(e^(3log x) - e^(2log x))`
Integrate the function:
`cos^3 xe^(log sinx)`
Integrate the function:
`sqrt((1-sqrtx)/(1+sqrtx))`
Integrate the function:
`(2+ sin 2x)/(1+ cos 2x) e^x`
Integrate the function:
`tan^(-1) sqrt((1-x)/(1+x))`
Evaluate `int(x^3+5x^2 + 4x + 1)/x^2 dx`
The anti derivative of `(sqrt(x) + 1/sqrt(x))` is equals:
`sqrt((10x^9 + 10^x log e^10)/(x^10 + 10^x)) dx` equals
`int (dx)/(sin^2x cos^2x) dx` equals
`int (dx)/(x(x^2 + 1))` equals
`int sqrt(1 + x^2) dx` is equal to
`int sqrt(x^2 - 8x + 7) dx` is equal to:-
What is anti derivative of `e^(2x)`
If the normal to the curve y(x) = `int_0^x(2t^2 - 15t + 10)dt` at a point (a, b) is parallel to the line x + 3y = –5, a > 1, then the value of |a + 6b| is equal to ______.
`int (dx)/sqrt(5x - 6 - x^2)` equals ______.
What is integration called because it reverses the operation of finding derivatives?
In \[\int f(x)\,dx\], what is \(f(x)\) called?
What is “an integral of \(f\)” ?
What is integration?
Which statement about indefinite integrals is correct?
For \(n\ne -1\), which is the correct integral of \(x^n\)?
Evaluate \[\int \sin x\,dx\]
Evaluate \[\int \text{cosec} x\cot x\,dx\]
Which expression is also an antiderivative of \[\frac{1}{\sqrt{1-x^2}}\]?
Evaluate \[\int \frac{dx}{1+x^2}\]
Evaluate \[\int \frac{1}{x}\,dx\]
