INTEGRATION BY ALGEBRAIC SUBSTITUTION
- is a method used to simplify integrals by substituting a part of the integrand with a new variable.
This transforms the integral into a simpler form, making it easier to evaluate.
Steps for Integration by Substitution
1. Choose a substitution:
Identify a portion of the integrand to substitute, and let u=g(x)u = g(x)u=g(x), where g(x)g(x)g(x)
is a function. Ensure its derivative, g (x)g'(x)g (x), is present (or can be adjusted) in the integrand.′ ′
2. Differentiate the substitution:
Compute du=g (x)dx = g'(x) dx=g (x)dx. This allows you to express dx in terms of du.′ ′
3. Rewrite the integral:
Substitute u=g(x)u = g(x)u=g(x) and replace dx with dug (x)\frac{du}{g'(x)} g (x)du′ ′ . Rewrite the
integrand entirely in terms of u.
4. Integrate in terms of u:
Solve the integral using the new variable u.
5. Back-substitute:
Replace u with the original expression g(x)g(x)g(x) to return the solution to its original variable.
Example 1:
Basic Substitution
Evaluate ∫2x(x2+1)3dx\int 2x (x^2 + 1)^3 dx∫2x(x2+1)3dx.
1. Choose a substitution: Let u=x2+1u = x^2 + 1u=x2+1. Then du=2xdx.
2. Rewrite the integral: Substituting u and du:
3. Integrate in terms of u:
4. Back-substitute: Replace u with x2+1x^2 + 1x2+1:
Example 2:
Trigonometric Substitution
Evaluate ∫dx1−x2\int \frac{dx}{\sqrt{1 - x^2}}∫1−x2dx.
1. Choose a substitution: Let x=sin(u)x = \sin(u)x=sin(u), so dx=cos(u) and 1−x2=cos(u).
2. Rewrite the integral:
3. Integrate in terms of u:
