-14fxdx where fx-x2- 1-x-2 and fx-3x-2-x-4 is

The correct option is D 613 ∫_1^4f(x)dx where, f(x)=x^2, 1≤ x lt;2 f(x)=3x, 2≤ x lt; 4 ∫_1^4f(x)dx=∫_1^2f(x)dx+∫_2^4f(x)dx ...

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Byju's Answer

Standard XII

Mathematics

Logarithmic Differentiation

∫14fxdx where...

Question

$\int_{1}^{4} f (x) d x$ where $f (x) = x^{2}$ ; $1 \leq x < 2$ and $f (x) = 3 x; 2 \leq x < 4$ is

\frac{7}{3}

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\frac{37}{3}

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\frac{23}{3}

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\frac{61}{3}

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Solution

The correct option is D $\frac{61}{3}$
$\int_{1}^{4} f (x) d x$
where, $f (x) = x^{2}, 1 \leq x < 2$
$f (x) = 3 x, 2 \leq x < 4$
$\int_{1}^{4} f (x) d x = \int_{1}^{2} f (x) d x + \int_{2}^{4} f (x) d x$
$= \int_{1}^{2} x^{2} d x + \int_{2}^{4} 3 x d x$
$= {[\frac{x^{3}}{3}]}_{1}^{2} + {[\frac{3 x^{2}}{2}]}_{2}^{4}$
$= \frac{8}{3} - \frac{1}{3} + \frac{3 (16)}{2} - \frac{3 (4)}{2}$
$= \frac{7}{3} + 18$
$= \frac{7 + 54}{3} = \frac{61}{3}$
Hence, option 'D' is correct.

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Similar questions

Q. (i)

\int_{1}^{4} f (x) d x, where f (x) = \{\begin{cases} 4 x + 3, if 1 \leq x \leq 2 \\ 3 x + 5, if 2 \leq x \leq 4 \end{cases}

(ii)

\int_{0}^{9} f (x) d x, where f (x) \{\begin{matrix} \sin x & , & 0 \leq x \leq π / 2 \\ 1 & , & π / 2 \leq x \leq 3 \\ e^{x - 3} & , & 3 \leq x \leq 9 \end{matrix}

(iii)

\int_{1}^{4} f (x) d x, where f (x) = \{\begin{matrix} 7 x + 3 & , & if 1 \leq x \leq 3 \\ 8 x & , & if 3 \leq x \leq 4 \end{matrix}

Q. Find the ranges of the following functions:

f (x) = 1 - x - x^{2}

f (x) = \frac{3 x + 7}{x - 8}

f (x) = x^{2} + 2 x + 8

f (x) = \frac{4 x - 7}{x - 3}

Let $f (x) = ⎧ ⎨ ⎩ \begin{matrix} ∣ ∣ x^{2} - 3 x ∣ ∣ + a, 0 \leq x < \frac{3}{2} - 2 x + 3 x \geq \frac{3}{2} \end{matrix}$ If f(x) has a local maximum at x =.

Q. If

\int_{- 1}^{4} f (x) d x = 4

and

\int_{2}^{4} [3 - f (x)] d x = 7 t h e n \int_{- 1}^{2} f (x) d x =

Q. If

f (x) = \frac{2}{x - 3}

g (x) = \frac{x - 3}{x + 4}

and

h (x) = - \frac{2 (2 x + 1)}{x^{2} + x - 12}

, then

lim x \to 3 [f (x) + g (x) + h (x)]

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∫41f(x)dx where f(x)=x2;1≤x<2 and f(x)=3x;2≤x<4 is

The correct option is D 613∫41f(x)dxwhere, f(x)=x2,1≤x<2f(x)=3x,2≤x<4∫41f(x)dx=∫21f(x)dx+∫42f(x)dx=∫21x2dx+∫423xdx=[x33]21+[3x22]42=83−13+3(16)2−3(4)2=73+18=7+543=613Hence, option 'D' is correct.

$\int_{1}^{4} f (x) d x$ where $f (x) = x^{2}$ ; $1 \leq x < 2$ and $f (x) = 3 x; 2 \leq x < 4$ is