Problem 9
In Exercises \(5-14,\) evaluate the integral. $$\int \frac{2 d x}{x^{2}+1}$$
Problem 9
In Exercises \(1-10\) , use separation of variables to solve the initial value problem. Indicate the domain over which the solution is valid. \(\frac{d y}{d x}=-2 x y^{2}\) and \(y=0.25\) when \(x=1\)
Problem 9
In Exercises \(1-10,\) find the indefinite integral. $$\int y \ln y d y$$
Problem 9
In Exercises \(7-12,\) use differentiation to verify the antiderivative formula. $$\int e^{2 x} d x=\frac{1}{2} e^{2 x}+C$$
Problem 10
In Exercises \(1-10\) , use separation of variables to solve the initial value problem. Indicate the domain over which the solution is valid. \(\frac{d y}{d x}=\frac{4 \sqrt{y} \ln x}{x}\) and \(y=1\) when \(x=e\)
Problem 10
In Exercises \(7-12,\) use differentiation to verify the antiderivative formula. $$\int 5^{x} d x=\frac{1}{\ln 5} 5^{x}+C$$
Problem 10
In Exercises \(1-10,\) find the general solution to the exact differential equation. $$\frac{d y}{d u}=4(\sin u)^{3}(\cos u)$$
Problem 10
In Exercises \(1-10,\) find the indefinite integral. $$\int t^{2} \ln t d t$$
Problem 10
In Exercises \(5-14,\) evaluate the integral. $$\int \frac{3 d x}{x^{2}+9}$$
Problem 11
In Exercises \(11-14\) , find the solution of the differential equation \(d y / d t=k y, k\) a constant, that satisfies the given conditions. $$k=1.5, \quad y(0)=100$$
