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.

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

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

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

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

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

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

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

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

1. Let $u = x^2$. 2. Find $du = 2x , dx$. 3. Rewrite the integral as $\int \cos(u) , du$. 4. Evaluate to get $\sin(u) + C$. 5. Back-substitute to get $\sin(x^2) + C$.

1. Let $u = x^2 + 1$. 2. Find $du = 2x , dx$. 3. Change limits: when $x=1$, $u=2$; when $x=2$, $u=5$. 4. Rewrite the integral as $\int_{2}^{5} \frac{1}{u^2} , du$. 5. Evaluate to get $\left[-\frac{1}{u}\right]_{2}^{5}$. 6. Calculate the definite integral: $(-\frac{1}{5}) - (-\frac{1}{2}) = \frac{3}{10}$.

1. Identify the inner function. 2. Choose $u$. 3. Find $du$. 4. Rewrite the integral in terms of $u$. 5. Evaluate the integral. 6. Back-substitute.

1. Identify the inner function. 2. Choose $u$. 3. Find $du$. 4. Change the limits of integration. 5. Rewrite the integral in terms of $u$. 6. Evaluate the integral.

1. Identify the inner function. 2. Choose $u$. 3. Find $du$. 4. Rewrite the integral in terms of $u$. 5. Evaluate the integral. 6. Back-substitute. 7. Evaluate with original limits.

Look for an inner function whose derivative is also present in the integral. Common choices include expressions within parentheses, under radicals, or in exponents.

Replace the chosen expression with 'u' and replace 'dx' with an expression involving 'du'. Ensure the entire integral is in terms of 'u'.

For indefinite integrals, substitute back the original expression in terms of 'x' for 'u'. For definite integrals, either change the limits of integration or substitute back before evaluating.

Isolate 'dx' by dividing by any constants that appear when finding 'du'. Include these constants when rewriting the integral.

Manipulate the equation for 'du' to match the remaining integrand by multiplying or dividing by constants. Adjust the integral accordingly.