Advertisements
Advertisements
प्रश्न
Advertisements
उत्तर
\[\int_0^\frac{\pi}{4} \left( a^2 \cos^2 x + b^2 \sin^2 x \right)dx\]
\[ = \int_0^\frac{\pi}{4} \left[ a^2 \left( \frac{1 + \cos2x}{2} \right) + b^2 \left( \frac{1 - \cos2x}{2} \right) \right]dx\]
\[ = \int_0^\frac{\pi}{4} \left[ \left( \frac{a^2 + b^2}{2} \right) + \left( \frac{a^2 - b^2}{2} \right)\cos2x \right]dx\]
\[ = \left( \frac{a^2 + b^2}{2} \right) \int_0^\frac{\pi}{4} dx + \left( \frac{a^2 - b^2}{2} \right) \int_0^\frac{\pi}{4} \cos2xdx\]
\[= \left.\left( \frac{a^2 + b^2}{2} \right) \times x\right|_0^\frac{\pi}{4} + \left.\left( \frac{a^2 - b^2}{2} \right) \times \frac{\sin2x}{2}\right|_0^\frac{\pi}{4} \]
\[ = \left( \frac{a^2 + b^2}{2} \right)\left( \frac{\pi}{4} - 0 \right) + \left( \frac{a^2 - b^2}{4} \right)\left( \sin\frac{\pi}{2} - \sin0 \right)\]
\[ = \left( \frac{a^2 + b^2}{2} \right)\frac{\pi}{4} + \left( \frac{a^2 - b^2}{4} \right)\left( 1 - 0 \right)\]
\[ = \left( a^2 + b^2 \right)\frac{\pi}{8} + \frac{1}{4}\left( a^2 - b^2 \right)\]
APPEARS IN
संबंधित प्रश्न
Evaluate the following integral:
If f(x) is a continuous function defined on [−a, a], then prove that
`int_0^1 sqrt((1 - "x")/(1 + "x")) "dx"`
Given that \[\int\limits_0^\infty \frac{x^2}{\left( x^2 + a^2 \right)\left( x^2 + b^2 \right)\left( x^2 + c^2 \right)} dx = \frac{\pi}{2\left( a + b \right)\left( b + c \right)\left( c + a \right)},\] the value of \[\int\limits_0^\infty \frac{dx}{\left( x^2 + 4 \right)\left( x^2 + 9 \right)},\]
The value of \[\int\limits_0^\pi \frac{1}{5 + 3 \cos x} dx\] is
\[\int\limits_0^\pi \cos 2x \log \sin x dx\]
\[\int\limits_2^3 \frac{\sqrt{x}}{\sqrt{5 - x} + \sqrt{x}} dx\]
\[\int\limits_0^{\pi/2} \frac{\sin^2 x}{\sin x + \cos x} dx\]
\[\int\limits_{- 1}^1 e^{2x} dx\]
\[\int\limits_0^2 \left( x^2 + 2 \right) dx\]
Using second fundamental theorem, evaluate the following:
`int_(-1)^1 (2x + 3)/(x^2 + 3x + 7) "d"x`
Using second fundamental theorem, evaluate the following:
`int_0^(pi/2) sqrt(1 + cos x) "d"x`
Evaluate the following:
`int_0^oo "e"^(- x/2) x^5 "d"x`
If `int (3"e"^x - 5"e"^-x)/(4"e"6x + 5"e"^-x)"d"x` = ax + b log |4ex + 5e –x| + C, then ______.
Verify the following:
`int (2x + 3)/(x^2 + 3x) "d"x = log|x^2 + 3x| + "C"`
