Percentage
A percentage is a number indicated as a fraction of 100. It is a dimensionless number often expressed using the symbol %.
Algebraic Expressions
In mathematics, an algebraic expression consists of constant(s), variable(s), and mathematical operators. It is made up of terms.
Numbers
Numbers are some measures used for counting. They can be compared one with another to know its position in the number line and determine which one is greater or lesser than the other.
Subtraction
Before we begin to understand the subtraction of algebraic expressions, we need to list out a few things that form the basis of algebra.
Addition
Before we begin to understand the addition of algebraic expressions, we need to list out a few things that form the basis of algebra.
![31. Explain in your own words how the picture proves the theorem. Which notation in the
proof is unfamiliar? What do you think it means?
Theorem 7 (Integration by Parts (IBP)) If u = u(x) and v = v(x) are functions
of x that have continuous derivatives, then
u dv = uv -
v du,
where dv = v'(x) dx and du = u'(x) dx.
It is important to select the functions u and dv carefully in the setup. Choose the
function u to be a function that will simplify by calculating the derivative. Choose the
function dv to be a function that will simplify by calculating the antiderivative.
For example, in the integral /
xe" dx, put u(x) = x and dv = e*dx. Then calculate
du = u'(x)dx = ldx and take an antiderivative v = ƒ dv = e² (letting the arbitrary
constant C = 0). Apply Theorem 7 to obtain
xe" dx =
u dv = uv -
v du = xe" –
e* dx = xe² – e* + C.
Sometimes IBP needs to be applied more than once. When applying IBP a second
time, make sure to make the same type of choices for u and v
The methods of IBP is also useful when integrating inverse transcendental functions,
such as the logarithm and the arcsine. If you need to integrate an inverse transcen-
dental function, try putting u equal to the inverse transcendental function and taking
a derivative to obtain an algebraic function.
32.
x²e3* dx
33.
x² cos x d.x
34.
x³ In 3x dx
35.
e5z cos(2x) dx
36.
| In(sin x) dæ
10
37.
In(4 + x²) dx
38.
a sec?(2r) dr](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6b09678e-74be-44ad-bc01-46561e6f0b6a%2Ff1152eb4-38a6-44df-bae0-5a4f71894f18%2Fo4iv5hh_processed.jpeg&w=3840&q=75)
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