Integration using Substitution: Fresh Take

Substitution for Indefinite Integrals

The Main Idea

Integration by substitution is a technique for evaluating integrals. It’s useful when the integrand is the result of a chain-rule derivative
The method involves changing variables to simplify the integral
Key form to recognize:
- $\int f(g(x))g'(x)dx$
Substitution Theorem:
- For $u = g(x)$ where $g'(x)$ is continuous: $\int f(g(x))g'(x)dx = \int f(u)du = F(u) + C = F(g(x)) + C$
Process:
- Choose $u = g(x)$ such that $g'(x)$ is part of the integrand
- Express the integral in terms of $u$
- Integrate with respect to $u$
- Substitute back to express the result in terms of $x$
Sometimes need to adjust by a constant factor when $du$ doesn’t match exactly
May need to express $x$ in terms of $u$ to eliminate all $x$ terms

Use substitution to find the antiderivative of $\displaystyle\int 3{x}^{2}{({x}^{3}-3)}^{2}dx.$

Let $u={x}^{3}-3.$

$\displaystyle\int 3{x}^{2}{({x}^{3}-3)}^{2}dx=\frac{1}{3}{({x}^{3}-3)}^{3}+C$

Use substitution to find the antiderivative of $\displaystyle\int {x}^{2}{({x}^{3}+5)}^{9}dx.$

Multiply the du equation by $\frac{1}{3}.$

$\frac{{({x}^{3}+5)}^{10}}{30}+C$

Use substitution to evaluate the integral $\displaystyle\int \frac{ \cos t}{{ \sin }^{2}t}dt.$

$-\dfrac{1}{ \sin t}+C$

Use substitution to evaluate the indefinite integral $\displaystyle\int { \cos }^{3}t \sin tdt.$

$-\dfrac{{ \cos }^{4}t}{4}+C$

The Main Idea

Substitution can be applied to definite integrals
The limits of integration must be adjusted when changing variables
The technique combines substitution with the Fundamental Theorem of Calculus
Substitution Theorem for Definite Integrals:
- If $u = g(x)$ and $g'(x)$ is continuous on $[a,b]$ , then: $\int_a^b f(g(x))g'(x)dx = \int_{g(a)}^{g(b)} f(u)du$
Process:
- Choose $u = g(x)$ as in indefinite integration
- Express the integrand in terms of $u$
- Change the limits of integration from $x$ to $u$ values
- Integrate with respect to $u$
- Evaluate the integral using the new limits
Alternative Approach:
- Perform substitution without changing limits
- Find the antiderivative in terms of $u$
- Substitute back to $x$ before evaluating at the original limits
Combining Techniques:
- Substitution may be used alongside other integration techniques
- Trigonometric identities might be needed before substitution

Use substitution to evaluate the definite integral ${\displaystyle\int }_{-1}^{0}y{(2{y}^{2}-3)}^{5}dy.$

$\frac{91}{3}$

Use substitution to evaluate ${\displaystyle\int }_{0}^{1}{x}^{2} \cos \left(\frac{\pi }{2}{x}^{3}\right)dx.$

$\frac{2}{3\pi }\approx 0.2122$