If f be a derivable function satisfying the relation \int_{0}^{t} f^{2} \left(\right. x \left.\right) dx + \int_{0}^{\pi / 4} t^{3} \left(tan\right)^{2} x \textrm{ } dx \textrm{ }\textrm{ } + \textrm{ }\textrm{ } t^{3} \int_{0}^{\pi / 4} \left(tan\right)^{4} x \textrm{ } dx = 2 \int_{0}^{t} xf \left(\right. x \left.\right) dx , \forall t \in R then

Engineering

Mathematics

Inequalities

Leibnitz Rule Derivative of Anti Derivatives

Question

If f be a derivable function satisfying the relation

$\int_{0}^{t} f^{2} (x) dx + \int_{0}^{π / 4} t^{3} \tan^{2} x dx + t^{3} \int_{0}^{π / 4} \tan^{4} x dx = 2 \int_{0}^{t} xf (x) dx, \forall t \in R$

then

f(1) = 1

f ²(1) + f ²(2) + f ²(3) + f ²(4) = 30

f ³(1) + f ³(2) = 10

f(1) + f(2) = 4

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$∵$ Given equation can be written as

$\int_{0}^{t} f^{2} (x) dx + \int_{0}^{π / 4} t^{3} (\tan^{2} x + \tan^{4} x) dx = 2 \int_{0}^{t} xf (x) dx$

$\Rightarrow \int_{0}^{t} f^{2} (x) dx + t^{3} \times \frac{1}{3} = 2 \int_{0}^{t} xf (x) dx$

Differentiation both sides, we get

f ²(t) + t² = 2t f(t)

⇒ (f(t) – t)² = 0 ⇒ f(t) = t

$∴$ f(x) = x