To simplify complex integrals by replacing a part of the integrand with a new variable, making the integral easier to evaluate.

The inner function is often a good candidate for substitution because its derivative is likely present in the integral due 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.

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.

Simplifying ensures that the resulting integral is in a form that can be easily evaluated using standard integration techniques.

$du$ 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 $u$.

Because the derivative of a constant is zero, any constant could be part of the antiderivative. $C$ represents all possible constants.

Method 1: Change the limits of integration to match the new variable. Method 2: Substitute back into the expression before evaluating.

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.

U-substitution allows you to navigate through complex integrals and make them much more manageable.

$\int f(g(x)) cdot g'(x) , dx = \int f(u) , du$, where $u = g(x)$ and $du = g'(x) , dx$.

$du = g'(x) , dx$

If the original limits are $x=a$ and $x=b$, then the new limits are $u=g(a)$ and $u=g(b)$.

$\int \cos(u) , du = \sin(u) + C$

$\int u^n , du = \frac{u^{n+1}}{n+1} + C$

$\int \frac{1}{u} , du = \ln|u| + C$

$\frac{d}{dx}x^n = nx^{n-1}$

$\frac{d}{dx}c = 0$

$\frac{d}{dx}[f(g(x))] = f'(g(x)) * g'(x)$

$\int 2xcos(x^2) dx = sin(x^2) + C$

A technique to simplify integrals by introducing a new variable, $u$, to transform the integral into a more recognizable form.

The function being integrated in an integral.

An integral without specified upper and lower limits, resulting in a function plus a constant of integration.

An integral with specified upper and lower limits, resulting in a numerical value.

The process of replacing the variable $u$ with its original expression in terms of $x$ after integration.

The upper and lower bounds of a definite integral, which specify the interval over which the integration is performed.

A function formed by substituting one function into another, typically in the form $f(g(x))$.

A constant term, denoted as $C$, added to the antiderivative of a function to represent the family of all possible antiderivatives.

A function whose derivative is equal to a given function. If $F'(x) = f(x)$, then $F(x)$ is an antiderivative of $f(x)$.

To find the antiderivative of the integrand and, for definite integrals, compute its value at the upper and lower limits of integration.