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Define indefinite integral.
An integral without specified bounds, resulting in a family of functions that differ by a constant.
What is the integration constant?
A constant term, denoted as 'C', added to the antiderivative to account for all possible antiderivatives.
Define antiderivative.
A function whose derivative is equal to a given function.
What is the reverse power rule?
A method for finding the antiderivative of power functions.
What is a family of functions?
A set of functions that share the same derivative but differ by a constant.
Define sums rule for antiderivatives.
The integral of a sum of functions is the sum of their individual integrals.
Define multiples rule for antiderivatives.
The integral of a constant times a function is the constant times the integral of the function.
What is the antiderivative of $\frac{1}{x}$?
$\ln|x| + C$
What is the antiderivative of $e^x$?
$e^x + C$
What is the antiderivative of $\sin(x)$?
$-\cos(x) + C$
How to solve $\int (3x^2 + 2x + 1) dx$?
Apply the sums rule: $\int 3x^2 dx + \int 2x dx + \int 1 dx$. Apply the reverse power rule to each term: $x^3 + x^2 + x + C$.
How to solve $\int 5\cos(x) dx$?
Apply the multiples rule: $5\int \cos(x) dx$. Integrate: $5\sin(x) + C$.
How to solve $\int (e^x - \sin(x)) dx$?
Apply the sums rule: $\int e^x dx - \int \sin(x) dx$. Integrate each term: $e^x + \cos(x) + C$.
How to solve $\int \frac{4}{x} dx$?
Apply the multiples rule: $4\int \frac{1}{x} dx$. Integrate: $4\ln|x| + C$.
How to solve $\int (\sqrt{x} + \frac{1}{x^2}) dx$?
Rewrite: $\int (x^{1/2} + x^{-2}) dx$. Apply the reverse power rule to each term: $\frac{2}{3}x^{3/2} - x^{-1} + C$.
Solve $\int (x^3 + 5) dx$.
$\int x^3 dx + \int 5 dx = \frac{x^4}{4} + 5x + C$
Solve $\int (2\sin(x) - 3\cos(x)) dx$.
$2\int \sin(x) dx - 3\int \cos(x) dx = -2\cos(x) - 3\sin(x) + C$
Solve $\int (\frac{1}{x^3} + e^x) dx$.
$\int x^{-3} dx + \int e^x dx = -\frac{1}{2x^2} + e^x + C$
Solve $\int (7x^6 - \frac{2}{x}) dx$.
$\int 7x^6 dx - \int \frac{2}{x} dx = x^7 - 2\ln|x| + C$
Solve $\int (\frac{5}{\sqrt{x}} - 4x^3) dx$.
$\int 5x^{-1/2} dx - \int 4x^3 dx = 10\sqrt{x} - x^4 + C$
Explain why '+C' is added to indefinite integrals.
Because the derivative of a constant is zero, any constant could be part of the original function. '+C' represents all possible constants.
Explain the reverse power rule.
To find the antiderivative of $x^n$, add 1 to the exponent and divide by the new exponent, then add the constant of integration, C.
Why is $n \neq -1$ in the reverse power rule?
Because if $n = -1$, the denominator becomes zero, making the expression undefined.
Explain the sums rule for antiderivatives.
The antiderivative of a sum of functions is the sum of their individual antiderivatives. This simplifies integration of complex expressions.
Explain the multiples rule for antiderivatives.
A constant factor can be moved outside the integral sign, simplifying the integration process.
Why is absolute value used in $\int \frac{1}{x} dx = \ln|x| + C$?
The natural logarithm is only defined for positive values. The absolute value ensures that the antiderivative is defined for all non-zero x.
Explain the relationship between derivatives and antiderivatives.
Antiderivatives are the inverse operation of derivatives. Finding the antiderivative reverses the process of differentiation.
Describe the process of finding the antiderivative of a polynomial.
Apply the reverse power rule to each term of the polynomial, then add the constant of integration, C.
Explain how to verify an antiderivative.
Differentiate the antiderivative. If the result is the original function, the antiderivative is correct.
Explain the difference between definite and indefinite integrals.
Definite integrals have upper and lower bounds and result in a numerical value. Indefinite integrals do not have bounds and result in a function plus a constant.