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 each of the following integral:
\[\int\limits_0^\infty \frac{1}{1 + e^x} dx\] equals
\[\int\limits_0^{\pi/2} \frac{1}{2 + \cos x} dx\] equals
The value of the integral \[\int\limits_0^\infty \frac{x}{\left( 1 + x \right)\left( 1 + x^2 \right)} dx\]
The value of \[\int\limits_0^1 \tan^{- 1} \left( \frac{2x - 1}{1 + x - x^2} \right) dx,\] is
\[\int\limits_0^1 \frac{1 - x}{1 + x} dx\]
Evaluate the following integrals :-
\[\int_2^4 \frac{x^2 + x}{\sqrt{2x + 1}}dx\]
\[\int\limits_0^{2\pi} \cos^7 x dx\]
\[\int\limits_0^\pi \frac{x}{a^2 - \cos^2 x} dx, a > 1\]
\[\int\limits_{\pi/6}^{\pi/2} \frac{\ cosec x \cot x}{1 + {cosec}^2 x} dx\]
\[\int\limits_1^4 \left( x^2 + x \right) dx\]
Evaluate the following using properties of definite integral:
`int_(- pi/2)^(pi/2) sin^2theta "d"theta`
Evaluate the following using properties of definite integral:
`int_0^(i/2) (sin^7x)/(sin^7x + cos^7x) "d"x`
Evaluate the following:
Γ(4)
Evaluate the following integrals as the limit of the sum:
`int_1^3 (2x + 3) "d"x`
Choose the correct alternative:
If n > 0, then Γ(n) is
Integrate `((2"a")/sqrt(x) - "b"/x^2 + 3"c"root(3)(x^2))` w.r.t. 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 ______.
Given `int "e"^"x" (("x" - 1)/("x"^2)) "dx" = "e"^"x" "f"("x") + "c"`. Then f(x) satisfying the equation is:
