Integration by substitution is a powerful technique for simplifying complex integrals. It involves replacing part of the integrand with a new variable, making the integral easier to solve. This method is especially useful for composite functions and expressions with linear terms.

Mastering substitution is crucial for tackling a wide range of integration problems. It's not just about following steps; it's about recognizing patterns and choosing the right substitution. With practice, you'll develop an intuition for when and how to apply this method effectively.

Substitution Method for Integration

Substitution for indefinite integrals

• Substitution simplifies the integration of composite functions $f(g(x))$ by introducing a new variable $u$
• Choose $u$ to represent part of the integrand that can be easily differentiated
  • For example, if the integrand contains $\sin(3x+1)$, let $u=3x+1$
• Express the entire integrand in terms of $u$, replacing $x$ with the inverse of the substitution
  • In the example, $\sin(3x+1)$ becomes $\sin(u)$
• Determine $du$ by differentiating the substitution for $u$
  • If $u=3x+1$, then $du=3dx$ or $dx=\frac{1}{3}du$
• Rewrite the integral using $u$ and $du$, multiplying by $\frac{du}{dx}$ to maintain equality
  • $\int \sin(3x+1) dx = \int \sin(u) \cdot \frac{1}{3}du$
• Integrate with respect to $u$ using known integration formulas
• Substitute the original expression for $u$ back into the result to obtain the antiderivative in terms of $x$
• This process is also known as change of variables in more advanced calculus

Substitution in definite integrals

• Substitution in definite integrals follows the same process as indefinite integrals with additional steps
• Express the limits of integration in terms of the substitution variable $u$
  • If $u=3x+1$ and the original limits are $x=0$ and $x=2$, the new limits are $u(0)=1$ and $u(2)=7$
• Evaluate the new integral with respect to $u$ using the transformed limits
  • $\int_0^2 \sin(3x+1) dx = \int_1^7 \sin(u) \cdot \frac{1}{3}du$
• The final result is a definite value, so there's no need to substitute back for $x$

Recognition of substitution-suitable integrands

• Substitution is useful when the integrand is a composite function or contains terms that can be simplified
• Powers of linear terms are good candidates for substitution
  • $\int (2x-3)^5 dx$ can be simplified by letting $u=2x-3$
• Trigonometric functions of linear terms are also suitable
  • $\int \cos(4x+2) dx$ can be simplified by letting $u=4x+2$
• Exponential functions of linear terms can be handled with substitution
  • $\int e^{-5x+1} dx$ can be simplified by letting $u=-5x+1$
• Square roots of linear terms are another common case for substitution
  • $\int \sqrt{x+7} dx$ can be simplified by letting $u=x+7$
• Choose a substitution that simplifies the integrand and makes it easier to integrate with respect to $u$

Advanced Integration Techniques

• Integration by parts is used when the integrand is a product of functions
• Inverse functions often require substitution or other advanced techniques for integration
• Substitution is a key method in solving certain types of differential equations