u-substitution with manipulation

U-substitution is a method used to simplify integrals by substituting a part of the integrand with a new variable, u, making the integral easier to solve.

Choosing u simplifies the integral, often transforming it into a basic form that can be easily integrated.

$$\frac{du}{dx} = 2\text{sec}^2(2x)$$ .

$$\text{ln}|u| + C$$ .

The chain rule is a formula for computing the derivative of the composition of two or more functions.

When using u-substitution, the chain rule helps to relate the derivative of u back to the original variable, allowing for proper integration.

$$-\frac{1}{3\text{cos}^3(2x)} + C$$ .

$$-\text{ln}|4-e^x| + C$$ .

$$-\frac{1}{(3x^2+1)^2} + C$$ .

$$\frac{1}{15}(3x-4)^5 + C$$ .