Advertisements
Advertisements
Question
\[\int\limits_0^{\pi/2} \frac{\cos x}{1 + \sin^2 x} dx\]
Advertisements
Solution
\[\int_0^\frac{\pi}{2} \frac{\cos x}{1 + \sin^2 x} d x\]
\[Let \sin x = t,\text{ then }\cos x dx = dt\]
\[\text{When }x \to 0 ; t \to 0\]
\[\text{And }x \to \frac{\pi}{2}; t \to 1\]
Therefore the integral becomes
\[ \int_0^1 \frac{dt}{1 + t^2}\]
\[ = \left[ \tan^{- 1} x \right]_0^1 \]
\[ = \frac{\pi}{4}\]
APPEARS IN
RELATED QUESTIONS
Evaluate the following integral:
Prove that:
\[\int\limits_0^1 \left\{ x \right\} dx,\] where {x} denotes the fractional part of x.
\[\int\limits_0^\pi \frac{1}{1 + \sin x} dx\] equals
If \[\int\limits_0^a \frac{1}{1 + 4 x^2} dx = \frac{\pi}{8},\] then a equals
The value of \[\int\limits_{- \pi/2}^{\pi/2} \left( x^3 + x \cos x + \tan^5 x + 1 \right) dx, \] is
\[\int\limits_0^{\pi/4} \cos^4 x \sin^3 x dx\]
\[\int\limits_1^3 \left| x^2 - 2x \right| dx\]
\[\int\limits_0^a \frac{\sqrt{x}}{\sqrt{x} + \sqrt{a - x}} dx\]
\[\int\limits_0^\pi x \sin x \cos^4 x dx\]
\[\int\limits_0^\pi \frac{x}{a^2 \cos^2 x + b^2 \sin^2 x} dx\]
\[\int\limits_0^{15} \left[ x^2 \right] dx\]
\[\int\limits_1^4 \left( x^2 + x \right) dx\]
Evaluate the following:
f(x) = `{{:("c"x",", 0 < x < 1),(0",", "otherwise"):}` Find 'c" if `int_0^1 "f"(x) "d"x` = 2
Evaluate the following using properties of definite integral:
`int_0^1 x/((1 - x)^(3/4)) "d"x`
Evaluate the following:
`int_0^oo "e"^(-mx) x^6 "d"x`
`int "e"^x ((1 - x)/(1 + x^2))^2 "d"x` is equal to ______.
Which integral has Constant C present?
