Let f(x) be a non-negative continuous functio | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Given,$\int_{\pi/4}^{\beta} f(x) dx = \beta \sin \beta + \frac{\pi}{4} \cos \beta + \sqrt{2} \beta$. On differentiating both sides with respect to

Vedclass · Accepted Answer

$\left( 1 - \frac{\pi}{4} + \sqrt{2} \right)$

Vedclass · Answer

(D) Given,$\int_{\pi/4}^{\beta} f(x) dx = \beta \sin \beta + \frac{\pi}{4} \cos \beta + \sqrt{2} \beta$. On differentiating both sides with respect to $\beta$ using the Leibniz integral rule,we get: $f(\beta) = \frac{d}{d\beta} \left( \beta \sin \beta + \frac{\pi}{4} \cos \beta + \sqrt{2} \beta \right)$. Applying the product rule and derivative rules: $f(\beta) = (1 \cdot \sin \beta + \beta \cos \beta) + \frac{\pi}{4} (-\sin \beta) + \sqrt{2}$. $f(\beta) = \sin \beta + \beta \cos \beta - \frac{\pi}{4} \sin \beta + \sqrt{2}$. To find $f\left( \frac{\pi}{2} \right)$,substitute $\beta = \frac{\pi}{2}$: $f\left( \frac{\pi}{2} \right) = \sin \left( \frac{\pi}{2} \right) + \frac{\pi}{2} \cos \left( \frac{\pi}{2} \right) - \frac{\pi}{4} \sin \left( \frac{\pi}{2} \right) + \sqrt{2}$. Since $\sin \left( \frac{\pi}{2} \right) = 1$ and $\cos \left( \frac{\pi}{2} \right) = 0$: $f\left( \frac{\pi}{2} \right) = 1 + \frac{\pi}{2}(0) - \frac{\pi}{4}(1) + \sqrt{2}$. $f\left( \frac{\pi}{2} \right) = 1 - \frac{\pi}{4} + \sqrt{2}$.

Let $f(x)$ be a non-negative continuous function such that the area bounded by the curve $y = f(x)$,$x$-axis and the ordinates $x = \frac{\pi}{4}$ and $x = \beta > \frac{\pi}{4}$ is $\left( \beta \sin \beta + \frac{\pi}{4} \cos \beta + \sqrt{2} \beta \right)$. Then $f\left( \frac{\pi}{2} \right)$ is

Similar Questions

If $x=a\left[\cos \theta+\log \left\{\tan \left(\frac{\theta}{2}\right)\right\}\right]$ and $y=a \sin \theta$,then $\frac{dy}{dx}$ is equal to

Which of the following is not a steroid hormone?

Which of the following statements is not correct regarding calcination?

Which of the following is the weakest base?

The $pH$ of $0.01 \ M$ solution of acetic acid is $5.0$. What are the values of $[H^{+}]$ and $K_a$ respectively?

Vedclass Test Series

Exam Paper Generator

Online Exam Module