Advertisements
Advertisements
Question
Evaluate : \[\int\frac{dx}{\sin^2 x \cos^2 x}\] .
Advertisements
Solution
Let I = \[\int\frac{dx}{\sin^2 x \cos^2 x}\]
Dividing the numerator and denominator by cos4 x, we get:
I = \[\int\frac{se c^2 x \cdot se c^2 x}{\tan^2 x}dx\]
\[\int\frac{\left( 1 + \tan^2 x \right) \cdot se c^2 x}{\tan^2 x}dx\]
Put tan x = t
⇒ \[se c^2 xdx = dt\]
∴ I = \[\int\frac{1 + t^2}{t^2}dt\] = \[\int1dt + \int\frac{1}{t^2}dt\]
⇒ I = t −\[\frac{1}{t}\] + C
⇒ I = tan x − cot x + C
∴ \[\int\frac{dx}{\sin^2 x \cos^2 x}\] = tan x − cot x + C
APPEARS IN
RELATED QUESTIONS
\[\int\limits_0^1 \tan^{- 1} \left( \frac{2x}{1 - x^2} \right) dx\]
\[\int\limits_0^\infty \frac{x}{\left( 1 + x \right)\left( 1 + x^2 \right)} dx\]
\[\int\limits_0^1 \left( \cos^{- 1} x \right)^2 dx\]
\[\int\limits_1^3 \left| x^2 - 4 \right| dx\]
\[\int\limits_0^{\pi/2} \frac{1}{2 \cos x + 4 \sin x} dx\]
\[\int\limits_0^2 \left( 2 x^2 + 3 \right) dx\]
Using second fundamental theorem, evaluate the following:
`int_1^2 (x "d"x)/(x^2 + 1)`
Using second fundamental theorem, evaluate the following:
`int_0^1 x"e"^(x^2) "d"x`
Evaluate the following:
`int_(-1)^1 "f"(x) "d"x` where f(x) = `{{:(x",", x ≥ 0),(-x",", x < 0):}`
Choose the correct alternative:
Using the factorial representation of the gamma function, which of the following is the solution for the gamma function Γ(n) when n = 8 is
Choose the correct alternative:
If n > 0, then Γ(n) is
Verify the following:
`int (x - 1)/(2x + 3) "d"x = x - log |(2x + 3)^2| + "C"`
`int x^9/(4x^2 + 1)^6 "d"x` is equal to ______.
Given `int "e"^"x" (("x" - 1)/("x"^2)) "dx" = "e"^"x" "f"("x") + "c"`. Then f(x) satisfying the equation is:
