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

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.