Chapter 6

In Exercises \(1-6,\) find the indefinite integral. $$\int\left(\cos x-3 x^{2}\right) d x$$

In Exercises \(1-10,\) find the indefinite integral. $$\int x \sin x d x$$

In Exercises \(1-4,\) find the values of \(A\) and \(B\) that complete the partial fraction decomposition. $$\frac{x-12}{x^{2}-4 x}=\frac{A}{x}+\frac{B}{x-4}$$

In Exercises \(1-10\) , use separation of variables to solve the initial value problem. Indicate the domain over which the solution is valid. \(\frac{d y}{d x}=\frac{x}{y} \quad\) and \(y=2\) when \(x=1\)

In Exercises \(1-10,\) find the general solution to the exact differential equation. \(\frac{d y}{d x}=5 x^{4}-\sec ^{2} x\)

In Exercises \(1-6,\) find the indefinite integral. $$\int x^{-2} d x$$

In Exercises \(1-10,\) find the indefinite integral. $$\int x e^{x} d x$$

In Exercises \(1-4,\) find the values of \(A\) and \(B\) that complete the partial fraction decomposition. $$\frac{2 x+16}{x^{2}+x-6}=\frac{A}{x+3}+\frac{B}{x-2}$$

In Exercises \(1-10\) , use separation of variables to solve the initial value problem. Indicate the domain over which the solution is valid. \(\frac{d y}{d x}=-\frac{x}{y} \quad\) and \(y=3\) when \(x=4\)

In Exercises \(1-10,\) find the general solution to the exact differential equation. $$\frac{d y}{d x}=\sec x \tan x-e^{x}$$

In Exercises \(1-6,\) find the indefinite integral. $$\int\left(t^{2}-\frac{1}{t^{2}}\right) d t$$

In Exercises \(1-10,\) find the indefinite integral. $$\int 3 t e^{2 t} d t$$

In Exercises \(1-4,\) find the values of \(A\) and \(B\) that complete the partial fraction decomposition. $$\frac{16-x}{x^{2}+3 x-10}=\frac{A}{x-2}+\frac{B}{x+5}$$

In Exercises \(1-10\) , use separation of variables to solve the initial value problem. Indicate the domain over which the solution is valid. \(\frac{d y}{d x}=\frac{y}{x} \quad\) and \(y=2\) when \(x=2\)

In Exercises \(1-10,\) find the general solution to the exact differential equation. $$\frac{d y}{d x}=\sin x-e^{-x}+8 x^{3}$$

In Exercises \(1-6,\) find the indefinite integral. $$\int \frac{d t}{t^{2}+1}$$

In Exercises \(1-10,\) find the indefinite integral. $$\int 2 t \cos (3 t) d t$$

In Exercises \(1-4,\) find the values of \(A\) and \(B\) that complete the partial fraction decomposition. $$\frac{3}{x^{2}-9}=\frac{A}{x-3}+\frac{B}{x+3}$$

In Exercises \(1-10\) , use separation of variables to solve the initial value problem. Indicate the domain over which the solution is valid. \(\frac{d y}{d x}=2 x y \quad\) and \(y=3\) when \(x=0\)

In Exercises \(1-10,\) find the general solution to the exact differential equation. $$\frac{d y}{d x}=\frac{1}{x}-\frac{1}{x^{2}}(x>0)$$

In Exercises \(1-6,\) find the indefinite integral. $$\int\left(3 x^{4}-2 x^{-3}+\sec ^{2} x\right) d x$$

In Exercises \(1-10,\) find the indefinite integral. $$\int x^{2} \cos x d x$$

In Exercises \(5-14,\) evaluate the integral. $$\int \frac{x-12}{x^{2}-4 x} d x$$

In Exercises \(1-10\) , use separation of variables to solve the initial value problem. Indicate the domain over which the solution is valid. \(\frac{d y}{d x}=(y+5)(x+2)\) and \(y=1\) when \(x=0\)

In Exercises \(1-10,\) find the general solution to the exact differential equation. $$\frac{d y}{d x}=5^{x} \ln 5+\frac{1}{x^{2}+1}$$

In Exercises \(1-6,\) find the indefinite integral. $$\int\left(2 e^{x}+\sec x \tan x-\sqrt{x}\right) d x$$

In Exercises \(5-14,\) evaluate the integral. $$\int \frac{2 x+16}{x^{2}+x-6} d x$$

In Exercises \(1-10\) , use separation of variables to solve the initial value problem. Indicate the domain over which the solution is valid. \(\frac{d y}{d x}=\cos ^{2} y \quad\) and \(y=0\) when \(x=0\)

In Exercises \(1-10,\) find the general solution to the exact differential equation. $$\frac{d y}{d x}=\frac{1}{\sqrt{1-x^{2}}}-\frac{1}{\sqrt{x}}$$

In Exercises \(7-12,\) use differentiation to verify the antiderivative formula. $$\int \csc ^{2} u d u=-\cot u+C$$

In Exercises \(1-10,\) find the indefinite integral. $$\int 3 x^{2} e^{2 x} d x$$

In Exercises \(5-14,\) evaluate the integral. $$\int \frac{2 x^{3}}{x^{2}-4} d x$$

In Exercises \(1-10\) , use separation of variables to solve the initial value problem. Indicate the domain over which the solution is valid. \(\frac{d y}{d x}=(\cos x) e^{y+\sin x} \quad\) and \(y=0\) when \(x=0\)

In Exercises \(1-10,\) find the general solution to the exact differential equation. $$\frac{d y}{d t}=3 t^{2} \cos \left(t^{3}\right)$$

In Exercises \(7-12,\) use differentiation to verify the antiderivative formula. $$\int \csc u \cot u=-\csc u+C$$

In Exercises \(1-10,\) find the indefinite integral. $$\int x^{2} \cos \left(\frac{x}{2}\right) d x$$

In Exercises \(5-14,\) evaluate the integral. $$\int \frac{x^{2}-6}{x^{2}-9} d x$$

In Exercises \(1-10\) , use separation of variables to solve the initial value problem. Indicate the domain over which the solution is valid. \(\frac{d y}{d x}=e^{x-y} \quad\) and \(y=2\) when \(x=0\)

In Exercises \(1-10,\) find the general solution to the exact differential equation. $$\frac{d y}{d t}=(\cos t) e^{\sin t}$$

In Exercises \(7-12,\) use differentiation to verify the antiderivative formula. $$\int e^{2 x} d x=\frac{1}{2} e^{2 x}+C$$

In Exercises \(1-10,\) find the indefinite integral. $$\int y \ln y d y$$

In Exercises \(5-14,\) evaluate the integral. $$\int \frac{2 d x}{x^{2}+1}$$

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 \(x=1\)

In Exercises \(1-10,\) find the general solution to the exact differential equation. $$\frac{d u}{d x}=\left(\sec ^{2} x^{5}\right)\left(5 x^{4}\right)$$

In Exercises \(7-12,\) use differentiation to verify the antiderivative formula. $$\int 5^{x} d x=\frac{1}{\ln 5} 5^{x}+C$$

In Exercises \(1-10,\) find the indefinite integral. $$\int t^{2} \ln t d t$$

In Exercises \(5-14,\) evaluate the integral. $$\int \frac{3 d x}{x^{2}+9}$$

In Exercises \(1-10\) , use separation of variables to solve the initial value problem. Indicate the domain over which the solution is valid. \(\frac{d y}{d x}=\frac{4 \sqrt{y} \ln x}{x}\) and \(y=1\) when \(x=e\)

In Exercises \(1-10,\) find the general solution to the exact differential equation. $$\frac{d y}{d u}=4(\sin u)^{3}(\cos u)$$

In Exercises \(7-12,\) use differentiation to verify the antiderivative formula. $$\int \frac{1}{1+u^{2}} d u=\tan ^{-1} u+C$$