_ ^ ^5 x ^4 x ;dx = - Vidyadip

MCQ

$\int_{}^{} {{{\sin }^5}x{{\cos }^4}x\;dx = } $

✓
$ - \frac{1}{5}{\cos ^5}x + \frac{2}{7}{\cos ^7}x - \frac{1}{9}{\cos ^9}x + c$
B
$\frac{1}{5}{\cos ^5}x + \frac{2}{7}{\cos ^7}x - \frac{1}{9}{\cos ^9}x + c$
C
$\frac{1}{5}{\cos ^5}x + \frac{2}{7}{\cos ^7}x + \frac{1}{9}{\cos ^9}x + c$
D
None of these

✓

Answer

Correct option: A.

$ - \frac{1}{5}{\cos ^5}x + \frac{2}{7}{\cos ^7}x - \frac{1}{9}{\cos ^9}x + c$

a
(a) Put $\cos x = t \Rightarrow - \sin x\,dx = dt,$ then
$\int_{}^{} {{{(1 - {{\cos }^2}x)}^2}.{{\cos }^4}x\sin x\,dx} = - \int_{}^{} {{{(1 - {t^2})}^2}.\,{t^4}dt} $
$ = - \frac{{{t^5}}}{5} + \frac{2}{7}{t^7} - \frac{1}{9}{t^9} + c = - \frac{{{{\cos }^5}x}}{5} + \frac{2}{7}{\cos ^7}x - \frac{1}{9}{\cos ^9}x + c$.
Aliter : By reduction formula.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

If $\int_a^b {{x^3}dx} = 0$ and $\int_a^b {{x^2}} dx = \frac{2}{3}$, then the value of $a$ and $b$ will be respectively

The length of the latus rectum of the parabola $9{x^2} - 6x + 36y + 19 = 0$

If $\alpha ,\beta ,\gamma $ are the cube roots of $p(p < 0)$, then for any $x,y$ and $z,\,\,\frac{{x\alpha + y\beta + z\gamma }}{{x\beta + y\gamma + z\alpha }} = $

Let $\begin{aligned} S _{ n }( x )=\log _{ a ^{1 / 2}} x +\log _{ a / 3} x +\log _{ a ^{1 / 6}} x \\+\log _{ a ^{1 / 11}} x +\log _{ a ^{1 / 18}} x +\log _{ a ^{1 / 27}} x +\ldots . \end{aligned}$

up to $n-$terms, where $a > 1$. If $S_{24}(x)=1093$ and $S _{12}(2 x )=265,$ then value of $a$ is equal to ..... .

If the components of $\overrightarrow{ a }=\alpha \hat{ i }+\beta \hat{ j }+\gamma \hat{ k }$ along and perpendiculer to $\overrightarrow{ b }=3 \hat{ i }+\hat{ j }-\hat{ k }$ respectively, are $\frac{16}{11}(3 \hat{ i }+\hat{ j }-\hat{ k })$ and $\frac{1}{11}(-4 \hat{ i }-5 \hat{ j }-17 \hat{ k })$, then $\alpha^2+\beta^2+\gamma^2$ is equal to :

If ${}^n{P_5} = 20.\;{}^n{P_3}$, then $n = $

$\frac{{{1^3} + {2^3} + {3^3} + {4^3} + {{........12}^3}}}{{{1^2} + {2^2} + {3^2} + {4^2} + {{.........12}^2}}} = $

Suppose a parabola $y=a x^2+b x+c$ has two $x$ intercepts, one positive and one negative, and its vertex is $(2,-2)$. Then, which of the following is true?

Differential coefficient of ${\cos ^{ - 1}}(\sqrt x )$ with respect to $\sqrt {(1 - x)} $ is

Let $f (x)$ = $\int\limits_2^x {\frac{{dt}}{{\sqrt {1 + {t^4}} }}} $ and $g$ be the inverse of $f.$ Then the value of $g’(0) $ is