Chapter 6: Problem 44
In Exercises \(25-46,\) use substitution to evaluate the integral. $$\int \frac{d x}{\sqrt{5 x+8}}$$
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In Exercises \(47-50,\) use integration by parts to establish the reduction formula. $$\int x^{n} e^{a x} d x=\frac{x^{n} e^{a x}}{a}-\frac{n}{a} \int x^{n-1} e^{a x} d x, \quad a \neq 0$$
Second-Order Differential Equations Find the specific solution to each of the following second-order initial value problems by first finding \(d y / d x\) and then finding \(y\) . (a) \(\frac{d^{2} y}{d x^{2}}=24 x^{2}-10\) when \(x=1, \frac{d y}{d x}=3\) and \(y=5\) (b) \(\frac{d^{2} y}{d x^{2}}=\cos x-\sin x when \)x=0, \frac{d y}{d x}=2\( and \)y=0\( (c) \)\frac{d^{2} y}{d x^{2}}=e^{x}-x\( when \)x=0, \frac{d y}{d x}=0\( and \)y=1$
In Exercises \(53-66,\) make a \(u\) -substitution and integrate from \(u(a)\) to \(u(b) .\) $$\int_{2}^{5} \frac{d x}{2 x-3}$$
Different Solutions? Consider the integral \(\int 2 \sec ^{2} x \tan x d x\) (a) Evaluate the integral using the substitution \(u=\tan x\) . (b) Evaluate the integral using the substitution \(u=\sec x\) . (c) Writing to Learn Explain why the different-looking answers in parts (a) and (b) are actually equivalent.
In Exercises \(47-52,\) use the given trigonometric identity to set up a \(u\) -substitution and then evaluate the indefinite integral. $$\int 4 \cos ^{2} x d x, \quad \cos 2 x=1-2 \cos ^{2} x$$
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