Advertisements
Advertisements
प्रश्न
\[\int\limits_0^{\pi/4} \sin 2x \sin 3x dx\]
Advertisements
उत्तर
\[Let, I = \int_0^\frac{\pi}{4} \sin2x \sin3x d x ..................(1)\]
\[ \Rightarrow I = \left[ - \sin2x\frac{\cos3x}{3} \right]_0^\frac{\pi}{4} + \int_0^\frac{\pi}{4} 2\cos2x\frac{\cos3x}{3}dx\]
\[ \Rightarrow I = \left[ - \sin2x\frac{\cos3x}{3} \right]_0^\frac{\pi}{4} + \frac{2}{3} \left[ \cos2x\frac{\sin3x}{3} \right]_0^\frac{\pi}{4} + \frac{4}{9} \int_0^\frac{\pi}{4} \sin2x \sin3x d x\]
\[ \Rightarrow I = \left[ - \sin2x\frac{\cos3x}{3} \right]_0^\frac{\pi}{4} + \frac{2}{3} \left[ \cos2x\frac{\sin3x}{3} \right]_0^\frac{\pi}{4} + \frac{4}{9}I ..............\left[From (1) \right]\]
\[ \Rightarrow \frac{5}{9}I = \left[ - \sin2x\frac{\cos3x}{3} \right]_0^\frac{\pi}{4} + \frac{2}{3} \left[ \cos2x\frac{\sin3x}{3} \right]_0^\frac{\pi}{4} \]
\[ \Rightarrow \frac{5}{9}I = \frac{1}{3\sqrt{2}} + 0\]
\[ \Rightarrow \frac{5}{9}I = \frac{1}{3\sqrt{2}}\]
\[ \therefore I = \frac{3}{5\sqrt{2}}\]
APPEARS IN
संबंधित प्रश्न
\[\int\limits_{\pi/4}^{\pi/2} \cot x\ dx\]
Evaluate each of the following integral:
\[\int\limits_0^1 \left\{ x \right\} dx,\] where {x} denotes the fractional part of x.
If \[\left[ \cdot \right] and \left\{ \cdot \right\}\] denote respectively the greatest integer and fractional part functions respectively, evaluate the following integrals:
If \[I_{10} = \int\limits_0^{\pi/2} x^{10} \sin x\ dx,\] then the value of I10 + 90I8 is
The value of \[\int\limits_0^\pi \frac{1}{5 + 3 \cos x} dx\] is
\[\int\limits_0^{\pi/4} \cos^4 x \sin^3 x dx\]
Evaluate the following integrals :-
\[\int_2^4 \frac{x^2 + x}{\sqrt{2x + 1}}dx\]
\[\int\limits_0^1 x \left( \tan^{- 1} x \right)^2 dx\]
\[\int\limits_0^{\pi/4} \tan^4 x dx\]
\[\int\limits_0^{\pi/2} \frac{1}{2 \cos x + 4 \sin x} dx\]
Using second fundamental theorem, evaluate the following:
`int_0^1 "e"^(2x) "d"x`
Using second fundamental theorem, evaluate the following:
`int_0^1 x"e"^(x^2) "d"x`
Evaluate the following:
`int_1^4` f(x) dx where f(x) = `{{:(4x + 3",", 1 ≤ x ≤ 2),(3x + 5",", 2 < x ≤ 4):}`
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 integrals as the limit of the sum:
`int_1^3 (2x + 3) "d"x`
