SATHEE: Indefinite Integration 3 Question 6

Indefinite Integration 3 Question 6

6. If $f (x)$ is differentiable and $\int_{0}^{t^{2}} x f (x) d x = \frac{2}{5} t^{5}$ , then $f \frac{4}{25}$ equals

(2004, 1M)

(a) $\frac{2}{5}$

(b) $- \frac{5}{2}$

(d) $\frac{5}{2}$

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Answer:

Correct Answer: 6. (a)

Solution:

Here, $\int_{0}^{t^{2}} x f (x) d x = \frac{2}{5} t^{5}$

Using Newton Leibnitz’s formula, differentiating both sides, we get

$\begin{array}{lll} t^{2} f (t^{2}) \frac{d}{d t} (t^{2}) - 0 \cdot f (0) \frac{d}{d t} (0) = 2 t^{4} \\ \Rightarrow & t^{2} f (t^{2}) 2 t = 2 t^{4} \Rightarrow f (t^{2}) = t \\ ∴ & f \frac{4}{25} = - \frac{2}{5} & putting t = \frac{2}{5} \\ \Rightarrow & f \frac{4}{25} = \frac{2}{5} \end{array}$

Indefinite Integration 3 Question 6

6. If $f (x)$ is differentiable and $\int_{0}^{t^{2}} x f (x) d x = \frac{2}{5} t^{5}$ , then $f \frac{4}{25}$ equals

Answer:

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Indefinite Integration 3 Question 6

6. If f(x) is differentiable and ∫0t2xf(x)dx=25t5, then f425 equals

Answer:

Engineering Preparation

Medical Preparation

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Other Resources

Syllabus

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Sample Mock Test

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NCERT Exercise Video Solution

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6. If $f (x)$ is differentiable and $\int_{0}^{t^{2}} x f (x) d x = \frac{2}{5} t^{5}$ , then $f \frac{4}{25}$ equals