91Ó°ÊÓ

In Exercises 35–42, find the particular solution that satisfies the differential equation and the initial condition. $$ g^{\prime}(x)=4 x^{2}, g(-1)=3 $$

Finding an Indefinite Integral In Exercises \(33-42,\) find the indefinite integral. $$ \int \csc ^{2}\left(\frac{x}{2}\right) d x $$

Using Properties of Definite Integrals In Exercises \(33-40\) , evaluate the integral using the following values. $$ \int_{2}^{4} x^{3} d x=60, \quad \int_{2}^{4} x d x=6, \quad \int_{2}^{4} d x=2 $$ $$ \int_{2}^{4} 25 d x $$

In Exercises 35–42, find the particular solution that satisfies the differential equation and the initial condition. $$ h^{\prime}(t)=8 t^{3}+5, h(1)=-4 $$

Area Use Simpson's Rule with \(n=14\) to approximate the area of the region bounded by the graphs of \(y=\sqrt{x} \cos x\) \(y=0, x=0,\) and \(x=\pi / 2\)

Using Properties of Definite Integrals In Exercises \(33-40\) , evaluate the integral using the following values. $$ \int_{2}^{4} x^{3} d x=60, \quad \int_{2}^{4} x d x=6, \quad \int_{2}^{4} d x=2 $$ $$ \int_{2}^{4}(x-9) d x $$

Finding an Indefinite Integral In Exercises \(33-42,\) find the indefinite integral. $$ \int \frac{1}{\theta^{2}} \cos \frac{1}{\theta} d \theta $$

Finding an Indefinite Integral In Exercises \(33-42,\) find the indefinite integral. $$ \int x \sin x^{2} d x $$

Using Properties of Definite Integrals In Exercises \(33-40\) , evaluate the integral using the following values. $$ \int_{2}^{4} x^{3} d x=60, \quad \int_{2}^{4} x d x=6, \quad \int_{2}^{4} d x=2 $$ $$ \int_{2}^{4}\left(x^{3}+4\right) d x $$

Finding a Limit In Exercises \(37-42\) , find a formula for the sum of \(n\) terms. Use the formula to find the limit as \(n \rightarrow \infty\) $$ \lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(\frac{3 i}{n}\right)\left(\frac{3}{n}\right) $$