Let y = f ( x ) be a differentiable function satisfying 2 x f ( t ) ? d t + 2 = x 2 2 + x 2 t 2 f ( t ) ? d t then 4 4 f ( x ) + x 9 - x 3 + x + 1 cos 2 x ? d x =

Question

Let $y = f (x)$ be a differentiable function satisfying $\int_{2}^{x} f (t) d t + 2 = \frac{x^{2}}{2} + \int_{x}^{2} t^{2} f (t) d t$ then $\int_{- \frac{π}{4}}^{\frac{π}{4}} \frac{f (x) + x^{9} - x^{3} + x + 1}{\cos^{2} x} d x =________$

Smart Achievers · Accepted Answer

(1)

$By differentiating w.r.t. x, we get (1 + x^{2}) f (x) = x \Rightarrow f (x) is odd$

$Required integral = \int_{- π / 4}^{π / 4} \sec^{2} x d x = 2$

Solution

Q.
Let $y = f (x)$ be a differentiable function satisfying $\int_{2}^{x} f (t) d t + 2 = \frac{x^{2}}{2} + \int_{x}^{2} t^{2} f (t) d t$ then $\int_{- \frac{π}{4}}^{\frac{π}{4}} \frac{f (x) + x^{9} - x^{3} + x + 1}{\cos^{2} x} d x =________$

1 2

2 3

3 4

4 0

Ans.
(1)

$By differentiating w.r.t. x, we get (1 + x^{2}) f (x) = x \Rightarrow f (x) is odd$

$Required integral = \int_{- π / 4}^{π / 4} \sec^{2} x d x = 2$

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Solution

Q. Let y=f(x) be a differentiable function satisfying ∫2xf(t) dt+2=x22+∫x2t2f(t) dt then ∫-π4π4f(x)+x9-x3+x+1cos2x dx=________

1 2

2 3

3 4

4 0

Ans. (1) By differentiating w.r.t. x, we get (1+x2)f(x)=x⇒f(x) is odd Required integral=∫-π/4π/4sec2x dx=2

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Q.
Let $y = f (x)$ be a differentiable function satisfying $\int_{2}^{x} f (t) d t + 2 = \frac{x^{2}}{2} + \int_{x}^{2} t^{2} f (t) d t$ then $\int_{- \frac{π}{4}}^{\frac{π}{4}} \frac{f (x) + x^{9} - x^{3} + x + 1}{\cos^{2} x} d x =________$

Ans.
(1)

$By differentiating w.r.t. x, we get (1 + x^{2}) f (x) = x \Rightarrow f (x) is odd$

$Required integral = \int_{- π / 4}^{π / 4} \sec^{2} x d x = 2$