91Ó°ÊÓ

Find \(d y / d x\) in Exercises \(31-36\) $$ y=\int_{0}^{x} \sqrt{1+t^{2}} d t $$

In Exercises \(29-32,\) graph each function \(f(x)\) over the given interval. Partition the interval into four subintervals of equal length. Then add to your sketch the rectangles associated with the Riemann sum \(\Sigma_{k=1}^{4} f\left(c_{k}\right) \Delta x_{k},\) given that \(c_{k}\) is the (a) left-hand endpoint, (b) right- hand endpoint, (c) midpoint of the \(k\) th subinterval. (Make a separate sketch for each set of rectangles.) $$ f(x)=\sin x, \quad[-\pi, \pi] $$

Find \(d y / d x\) in Exercises \(31-36\) $$ y=\int_{1}^{x} \frac{1}{t} d t, \quad x>0 $$

Evaluate the integrals in Exercises \(13-48\) . $$ \int x^{1 / 3} \sin \left(x^{4 / 3}-8\right) d x $$

In Exercises \(29-32,\) graph each function \(f(x)\) over the given interval. Partition the interval into four subintervals of equal length. Then add to your sketch the rectangles associated with the Riemann sum \(\Sigma_{k=1}^{4} f\left(c_{k}\right) \Delta x_{k},\) given that \(c_{k}\) is the (a) left-hand endpoint, (b) right- hand endpoint, (c) midpoint of the \(k\) th subinterval. (Make a separate sketch for each set of rectangles.) $$ f(x)=\sin x+1, \quad[-\pi, \pi] $$

Evaluate the integrals in Exercises \(13-48\) . $$ \int \sec \left(v+\frac{\pi}{2}\right) \tan \left(v+\frac{\pi}{2}\right) d v $$

Evaluate the integrals in Exercises \(13-48\) . $$ \int \csc \left(\frac{v-\pi}{2}\right) \cot \left(\frac{v-\pi}{2}\right) d v $$

Find \(d y / d x\) in Exercises \(31-36\) $$ y=\int_{0}^{x^{2}} \cos \sqrt{t} d t $$

Find \(d y / d x\) in Exercises \(31-36\) $$ y=\int_{0}^{\sin x} \frac{d t}{\sqrt{1-t^{2}}}, \quad|x|<\frac{\pi}{2} $$

Evaluate the integrals in Exercises \(13-48\) . $$ \int \frac{\sin (2 t+1)}{\cos ^{2}(2 t+1)} d t $$