Advertisements
Advertisements
Question
\[\int\limits_0^{\pi/4} \cos^4 x \sin^3 x dx\]
Advertisements
Solution
\[\int_0^\frac{\pi}{4} \cos^4 x \sin^3 x d x\]
\[ = \int_0^\frac{\pi}{4} \cos^4 x \sin x \left( 1 - \cos^2 x \right) dx\]
\[ = \int_0^\frac{\pi}{4} \cos^4 x \sin x dx - \int_0^\frac{\pi}{4} \cos^6 x \sin x dx\]
\[ = - \left[ \frac{\cos^5 x}{5} \right]_0^\frac{\pi}{4} + \left[ \frac{\cos^7 x}{7} \right]_0^\frac{\pi}{4} \]
\[ = \frac{- 1}{20\sqrt{2}} + \frac{1}{5} + \frac{1}{56\sqrt{2}} - \frac{1}{7}\]
\[ = \frac{- \sqrt{2}}{40} + \frac{2}{35} + \frac{\sqrt{2}}{112}\]
\[ = \frac{2}{35} - \frac{9\sqrt{2}}{560}\]
APPEARS IN
RELATED QUESTIONS
Evaluate the following integral:
If \[f\left( x \right) = \int_0^x t\sin tdt\], the write the value of \[f'\left( x \right)\]
If f (a + b − x) = f (x), then \[\int\limits_a^b\] x f (x) dx is equal to
\[\int\limits_0^{\pi/2} \frac{\sin x}{\sqrt{1 + \cos x}} dx\]
\[\int\limits_1^2 \frac{1}{x^2} e^{- 1/x} dx\]
\[\int\limits_1^2 \frac{x + 3}{x\left( x + 2 \right)} dx\]
\[\int\limits_0^{\pi/2} \frac{1}{1 + \cot^7 x} dx\]
\[\int\limits_0^{\pi/2} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x} dx\]
\[\int\limits_0^\pi \cos 2x \log \sin x dx\]
\[\int\limits_0^\pi \frac{x}{a^2 - \cos^2 x} dx, a > 1\]
\[\int\limits_2^3 \frac{\sqrt{x}}{\sqrt{5 - x} + \sqrt{x}} dx\]
Find : `∫_a^b logx/x` dx
Using second fundamental theorem, evaluate the following:
`int_0^1 x"e"^(x^2) "d"x`
Using second fundamental theorem, evaluate the following:
`int_0^(pi/2) sqrt(1 + cos x) "d"x`
Using second fundamental theorem, evaluate the following:
`int_1^2 (x - 1)/x^2 "d"x`
Evaluate the following:
Γ(4)
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
Verify the following:
`int (x - 1)/(2x + 3) "d"x = x - log |(2x + 3)^2| + "C"`
The value of `int_2^3 x/(x^2 + 1)`dx is ______.
Geometrically, a definite integral is interpreted as what?
What is the meaning of a definite integral?
