Advertisements
Advertisements
प्रश्न
Evaluate the following using properties of definite integral:
`int_0^(i/2) (sin^7x)/(sin^7x + cos^7x) "d"x`
Advertisements
उत्तर
Using the property
`int_0^"a" "f"(x) "d"x = int_0^"a" "f"("a" - x) "d"x`
Let I = `int_0^(pi/2) (sin^7x)/(sin^7x + cos^7x) "d"x` ........(1)
I = `int_0^(pi/2) (sin^7(pi/2 - x))/(sin^7(pi/2 - x) + cos^7(pi/2 - x)) "d"x`
I = `int_0^(pi/2) (cos^7x)/(cos^7x + sin^x) "d"x` .........(2)
Adding (1) and (2)
I + I = `int_0^(pi/2) (sin^7x + cos^7x)/(sin^7x + cos^7x "d"x`
2I `int_0^(pi/2) "d"x`
2I = `[x]_0^(pi/2) = [pi/2 - 0]`
2I = `pi/2`
⇒ I = `pi/4`
APPEARS IN
संबंधित प्रश्न
If \[\left[ \cdot \right] and \left\{ \cdot \right\}\] denote respectively the greatest integer and fractional part functions respectively, evaluate the following integrals:
\[\int\limits_0^{2a} f\left( x \right) dx\] is equal to
Evaluate : \[\int\limits_0^{2\pi} \cos^5 x dx\] .
\[\int\limits_1^5 \frac{x}{\sqrt{2x - 1}} dx\]
\[\int\limits_0^a \frac{\sqrt{x}}{\sqrt{x} + \sqrt{a - x}} dx\]
Using second fundamental theorem, evaluate the following:
`int_0^3 ("e"^x "d"x)/(1 + "e"^x)`
Evaluate the following integrals as the limit of the sum:
`int_1^3 x "d"x`
