The derivative of f x- x 2 x 31 loge t dt- x-0- isa 13 ln xb 13 ln x-12 ln xc ln x-1 xx-1d 3 x 2 ln x

(c) (ln x) minus;1 x (x minus; 1) Using Newton Leibnitz formula f'(x)=1logex3(3x2) #8722;1logex2(2x)=3x23lnx #8722;2x2lnx=x2lnx #8722;xlnx=1lnxx(x ...

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Byju's Answer

Standard XII

Mathematics

Integration of Irrational Algebraic Fractions - 1

The derivativ...

Question

The derivative of $f (x) = \int_{x^{2}}^{x^{3}} \frac{1}{\log_{e} t} d t, (x > 0),$ is

(a) $\frac{1}{3 \ln x}$

(b) $\frac{1}{3 \ln x} - \frac{1}{2 \ln x}$

(c) (ln x)⁻¹ x (x − 1)

(d) $\frac{3 x^{2}}{\ln x}$

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Solution

(c) (ln x)⁻¹ x (x − 1)

Using Newton Leibnitz formula

$\begin{matrix} f^{'} (x) = \frac{1}{\log_{e} x^{3}} (3 x^{2}) - \frac{1}{\log_{e} x^{2}} (2 x) \\ = \frac{3 x^{2}}{3 \ln x} - \frac{2 x}{2 \ln x} \\ = \frac{x^{2}}{\ln x} - \frac{x}{\ln x} \\ = \frac{1}{\ln x} x (x - 1) \\ = {(\ln x)}^{- 1} x (x - 1) \end{matrix}$

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The derivative of fx=∫x2x31loge t dt, x>0, is (a) 13 ln x (b) 13 ln x-12 ln x (c) (ln x)−1 x (x − 1) (d) 3x2ln x

(c) (ln x)−1 x (x − 1) Using Newton Leibnitz formula f'(x)=1logex3(3x2)−1logex2(2x)=3x23lnx−2x2lnx=x2lnx−xlnx=1lnxx(x−1)=(lnx)−1x(x−1)

The derivative of $f (x) = \int_{x^{2}}^{x^{3}} \frac{1}{\log_{e} t} d t, (x > 0),$ is

(a) $\frac{1}{3 \ln x}$

(b) $\frac{1}{3 \ln x} - \frac{1}{2 \ln x}$

(c) (ln x)⁻¹ x (x − 1)

(d) $\frac{3 x^{2}}{\ln x}$