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What is the general form for u-substitution?
$\int f(g(x)) cdot g'(x) , dx = \int f(u) , du$, where $u = g(x)$ and $du = g'(x) , dx$.
If $u = g(x)$, what is $du$?
$du = g'(x) , dx$
How to transform limits of integration with u-substitution?
If the original limits are $x=a$ and $x=b$, then the new limits are $u=g(a)$ and $u=g(b)$.
What is the integral of $\cos(u)$ with respect to $u$?
$\int \cos(u) , du = \sin(u) + C$
What is the integral of $u^n$ with respect to $u$, where $n \neq -1$?
$\int u^n , du = \frac{u^{n+1}}{n+1} + C$
What is the integral of $\frac{1}{u}$ with respect to $u$?
$\int \frac{1}{u} , du = \ln|u| + C$
What is the derivative of $x^n$?
$\frac{d}{dx}x^n = nx^{n-1}$
What is the derivative of a constant?
$\frac{d}{dx}c = 0$
What is the chain rule?
$\frac{d}{dx}[f(g(x))] = f'(g(x)) * g'(x)$
What is the integral of $2xcos(x^2)$?
$\int 2xcos(x^2) dx = sin(x^2) + C$
Explain the goal of u-substitution.
To simplify complex integrals by replacing a part of the integrand with a new variable, making the integral easier to evaluate.
Why is identifying the inner function important in u-substitution?
The inner function is often a good candidate for substitution because its derivative is likely present in the integral due to the chain rule.
How does u-substitution relate to the chain rule?
U-substitution can be thought of as the 'reverse' of the chain rule because it involves identifying an expression whose derivative is in the integral.
Explain why changing the limits of integration is useful in definite integrals.
Changing the limits allows you to evaluate the integral directly in terms of $u$ without having to back-substitute, which can save time and reduce errors.
Describe the importance of simplifying the integral after substitution.
Simplifying ensures that the resulting integral is in a form that can be easily evaluated using standard integration techniques.
Explain the role of $du$ in u-substitution.
$du$ represents the derivative of the substituted variable $u$ and is used to replace $dx$ in the integral, ensuring the integral is entirely in terms of $u$.
Why do we add a constant $C$ in indefinite integrals?
Because the derivative of a constant is zero, any constant could be part of the antiderivative. $C$ represents all possible constants.
Explain the two methods for definite integrals.
Method 1: Change the limits of integration to match the new variable. Method 2: Substitute back into the expression before evaluating.
Explain the relationship between derivatives and integrals.
Integration is the reverse process of differentiation. The integral finds the area under the curve, while the derivative finds the slope of the tangent line.
Why is u-substitution a valuable tool?
U-substitution allows you to navigate through complex integrals and make them much more manageable.
What is u-substitution?
A technique to simplify integrals by introducing a new variable, $u$, to transform the integral into a more recognizable form.
What is an integrand?
The function being integrated in an integral.
What is an indefinite integral?
An integral without specified upper and lower limits, resulting in a function plus a constant of integration.
What is a definite integral?
An integral with specified upper and lower limits, resulting in a numerical value.
What is back-substitution?
The process of replacing the variable $u$ with its original expression in terms of $x$ after integration.
Define limits of integration.
The upper and lower bounds of a definite integral, which specify the interval over which the integration is performed.
What is a composite function?
A function formed by substituting one function into another, typically in the form $f(g(x))$.
What is the constant of integration?
A constant term, denoted as $C$, added to the antiderivative of a function to represent the family of all possible antiderivatives.
Define antiderivative.
A function whose derivative is equal to a given function. If $F'(x) = f(x)$, then $F(x)$ is an antiderivative of $f(x)$.
What does it mean to evaluate an integral?
To find the antiderivative of the integrand and, for definite integrals, compute its value at the upper and lower limits of integration.