⧼exchistory⧽
7 exercise(s) shown, 0 hidden
BBy Bot
Nov 03'24
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[/math]
Integrate each of the following.
- [math]\int x \cos x \; dx[/math]
- [math]\int (x+2) e^{2x} \; dx[/math]
- [math]\int \arctan x \; dx[/math]
- [math]\int x^2\sin 7x \; dx[/math]
- [math]\int x^3 \ln x \; dx[/math]
- [math]\int (x^3-7x+2) \sin x \; dx[/math]
- [math]\int e^{3x} \sin 2x \; dx[/math]
- [math]\int \sec^3x \; dx[/math]
- [math]\int x^2\ln(x+1) \; dx[/math]
- [math]\int \ln(x^2+2) \; dx[/math].
BBy Bot
Nov 03'24
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[/math]
Do Example \ref{exam 7.1.3} again, first letting [math]u = \cos 3x[/math] and then [math]u_1 = \sin 3x[/math].
BBy Bot
Nov 03'24
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[/math]
Find formulas for
- [math]\int e^{\alpha x} \cos bx \; dx[/math]
- [math]\int e^{\alpha x} \sin bx \; dx[/math].
BBy Bot
Nov 03'24
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[/math]
Derive the recursion formula analogous to: For every integer [math]n \geq 2[/math],
[[math]]
\int \sin^nx \; dx =
-\frac{\sin^{n-1}x \cos x}n + \frac{n-1}n
\int \sin^{n-2} x \; dx
.
[[/math]]
BBy Bot
Nov 03'24
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[/math]
Evaluate
- [math]\int \cos^2x \; dx[/math]
- [math]\int \sin^2x \; dx[/math]
by the recursion formulas [see and Problem Exercise], and also using the trigonometric identities
[[math]]
\cos^2x = \frac12 (1+\cos2x)
,
[[/math]]
[[math]]
\sin^2x = \frac12 (1-\cos2x)
.
[[/math]]
Show that the results obtained are the same. (The preceding identities can be read off at once from two more basic ones:
[[math]]
1 = \cos^2x + \sin^2x
[[/math]]
[[math]]
\cos 2x = \cos^2x - \sin^2x
.)
[[/math]]
BBy Bot
Nov 03'24
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[/math]
Use integration by parts to find recursion formulas, expressing the given integral in terms of an integral with a lower power:
- Show that [math]\int x^ne^xdx = x^ne^x - n\int x^{n-1}e^xdx[/math].
- Show that [math]\int \sec^nx \; dx = \frac{\sec^{n-2}x \tan x}{n-1} + \frac{n-2}{n-1} \int \sec^{n-2} x \; dx[/math].
- Find a reduction formula, expressing [math]\int(\ln |ax + b|)^n dx[/math] in terms of [math]\int(\ln|ax + b|)^{n-1}dx[/math].
BBy Bot
Nov 03'24
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[/math]
Use the formulas derived in and in Problems Exercise and Exercise to find
- [math]\int x^5e^x \; dx[/math]
- [math]\int \sin^4x \; dx[/math]
- [math]\int \cos^35x\; dx[/math]
- [math]\int(\ln|3x+7|)^6 \; dx[/math]
- [math]\int_0^{\frac{\pi}2} \sin^3x \; dx[/math]
- [math]\int_0^1 x^3e^x \; dx[/math].