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Questions  >>  CBSE XII  >>  Math  >>  Integrals
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Q)

Evaluate : $\begin{align*}\int_4^9 \frac{\sqrt x}{(30 - x^{\frac{3}{2}})^2} dx \end{align*}$


$(A)\; -\frac{19}{99}$
$(B)\; \frac{19}{99}$
$(C)\; \frac{-19}{66}$
$(D)\;\frac{19}{66}$

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A)
Put $30 - x^\frac{3}{2} = t$
$\therefore -\frac{3}{2} \sqrt x dx = dt $
$ \begin{align*} \therefore \sqrt x dx = -\frac{2}{3} dt \end{align*} \; \; \; when\; x =4, t= 22$
$\therefore \begin{align*} I = -\frac{2}{3} \int _{22}^3 \frac{dt}{t^2} \end{align*} \; \; \; when \; x = 9, t=3$
$\begin{align*} =+ \frac{2}{3} \int_3^{22} \frac{dt}{t^2} \end{align*}$
$ = -\frac {2}{3} \begin{bmatrix} \frac{1}{t}\end{bmatrix}_3^{22}$
$\begin{align*} = -\frac{2}{3} \begin{bmatrix} \frac{1}{22} - \frac{1}{3} \end{bmatrix} &= -\frac{2}{3} \begin{bmatrix} -\frac{19}{66} \end{bmatrix} \\ &=\frac{19}{99} \end{align*}$
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