Calculo Integral Por Sustitucion

Integral substitution, also known as u-substitution or change of variables, is a powerful technique in calculus for simplifying complex integrals. This method allows you to transform an integral into a simpler form by substituting a new variable for part of the integrand.

What is Integral Substitution?

Integral substitution is a method of solving integrals by replacing a complicated part of the integrand with a simpler variable. The process involves:

Identifying a suitable substitution (u)
Expressing the differential (du) in terms of the original variable
Rewriting the integral in terms of u
Solving the simpler integral
Substituting back to the original variable

The general substitution formula is:

If f(x) = g(u) and du = f'(x)dx, then:

∫ g(u) f'(x) dx = ∫ g(u) du

This technique is particularly useful for integrals involving composite functions, logarithmic functions, and rational functions.

How to Use Substitution

Step 1: Choose the substitution

Select a substitution u that simplifies the integrand. Common choices include:

Trigonometric functions (u = sin x, u = e^x, etc.)
Composite functions (u = x² + 1)
Logarithmic arguments (u = ln x)

Step 2: Find du

Differentiate u with respect to x to find du. This will give you the relationship between dx and du.

Step 3: Rewrite the integral

Express the original integral in terms of u using the relationship from step 2.

Step 4: Solve the simpler integral

Integrate the rewritten expression with respect to u.

Step 5: Substitute back

Replace u with the original expression to get the final answer.

Remember to include the constant of integration (C) when solving indefinite integrals.

Example Problems

Example 1: Simple substitution

Find ∫ 2x e^(x²) dx

Solution:

Let u = x², du = 2x dx
Rewrite the integral: ∫ e^u du
Integrate: e^u + C
Substitute back: e^(x²) + C

Example 2: Composite function

Find ∫ cos(3x) dx

Solution:

Let u = 3x, du = 3 dx → dx = du/3
Rewrite the integral: (1/3) ∫ cos(u) du
Integrate: (1/3) sin(u) + C
Substitute back: (1/3) sin(3x) + C

Comparison of substitution results
Original Integral	Substitution	Result
∫ 2x e^(x²) dx	u = x²	e^(x²) + C
∫ cos(3x) dx	u = 3x	(1/3) sin(3x) + C

Common Mistakes

Forgetting to multiply by the derivative when substituting back
Choosing a substitution that doesn't simplify the integral
Omitting the constant of integration for indefinite integrals
Incorrectly differentiating to find du
Not checking the limits of integration when solving definite integrals

FAQ

When should I use substitution instead of other integration techniques?: Use substitution when the integrand is a composite function or when you can identify a part of the integrand that can be simplified by substitution.
Can substitution be used for all types of integrals?: No, substitution works best for integrals involving composite functions, logarithmic functions, and rational functions. Other techniques may be more appropriate for other types of integrals.
What if my substitution doesn't simplify the integral?: If your substitution doesn't simplify the integral, try a different substitution or consider using another integration technique.
How do I know if I've done the substitution correctly?: Check that you've correctly identified u and du, that you've rewritten the integral in terms of u, and that you've substituted back correctly to the original variable.
What if I can't find a substitution that works?: If you can't find a suitable substitution, consider using integration by parts, partial fractions, or other techniques.