U-substitution

U-substitution is a method used to simplify the process of integration by substituting a part of the integrand with a new variable, usually denoted as 'u', to make the integral easier to solve.

Use U-substitution when the integrand is a composite function or when a function and its derivative appear in the integral.

Identify a part of the integrand to substitute with 'u', typically a function whose derivative is also present in the integrand.

After substituting u, differentiate u with respect to x to find du, and then express dx in terms of du.

If u = g(x), then the integral becomes: $$\int f(g(x))g'(x)dx = \int f(u)du + C$$ .

Let u = 5x, then du = 5dx, or dx = du/5. The integral becomes $$\int e^u du = e^u + C = e^{5x} + C$$ .

Let u = \ln x, then du = \frac{1}{x}dx. The integral becomes $$\int u^6 du = \frac{1}{7}u^7 + C = \frac{1}{7}(\ln x)^7 + C$$ .

Let u = x^2, then du = 2xdx. The integral becomes $$\int \cos(u) du = \sin(u) + C = \sin(x^2) + C$$ .

Let u = 3 + x^2, then du = 2xdx. The integral becomes $$\int \frac{1}{\sqrt{u}} du = 2\sqrt{u} + C = 2\sqrt{3+x^2} + C$$ .

The constant C represents the family of antiderivatives, as indefinite integrals can differ by a constant.