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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 $\int (\frac{5}{\sqrt{x}} - 4x^3) dx$.
$\int 5x^{-1/2} dx - \int 4x^3 dx = 10\sqrt{x} - x^4 + C$
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 $\sin(x)$?
$-\cos(x) + C$
What is the indefinite integral notation?
$\int f(x) dx = F(x) + C$, where $F'(x) = f(x)$
What is the reverse power rule formula?
$\int x^n dx = \frac{x^{n+1}}{n+1} + C$, where $n \neq -1$
What is the sums rule formula for antiderivatives?
$\int [f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx$
What is the multiples rule formula for antiderivatives?
$\int c \cdot f(x) dx = c \int f(x) dx$
What is the formula for $\int \sin(x) dx$?
$-\cos(x) + C$
What is the formula for $\int \cos(x) dx$?
$\sin(x) + C$
What is the formula for $\int \sec^2(x) dx$?
$\tan(x) + C$
What is the formula for $\int \csc^2(x) dx$?
$-\cot(x) + C$
What is the formula for $\int \sec(x)\tan(x) dx$?
$\sec(x) + C$
What is the formula for $\int \csc(x)\cot(x) dx$?
$-\csc(x) + C$
What is the formula for $\int \frac{1}{x} dx$?
$\ln|x| + C$
What is the formula for $\int e^x dx$?
$e^x + C$