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squenshl
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How do I evaluate the integral [tex]\int_a^b[/tex] f(x4) dx = [tex]\int_y^z[/tex] f(u) dx/du du, where u=x4
y=a4
z=b4
y=a4
z=b4
squenshl said:How do I evaluate the integral [tex]\int_a^b[/tex] f(x4) dx = [tex]\int_y^z[/tex] f(u) dx/du du, where u=x4
y=a4
z=b4
jgens said:Why would you use that method to evaluate the integral? The integral you have posted can easily be evaluated using the power rule for integration.
Integral substitution is a technique used in calculus to simplify integrals by replacing a variable with another variable or expression. This allows for the use of known integration rules and makes it easier to solve the integral.
To evaluate an integral using substitution, you need to identify a function within the integral that can be substituted with a new variable. This new variable should be chosen in a way that simplifies the integral. Then, use the appropriate substitution rule to rewrite the integral in terms of the new variable and solve.
The general formula for integral substitution is ∫f(g(x))g'(x)dx = ∫f(u)du, where g(x) is the substituted variable and u is the new variable. This formula is known as the substitution rule.
Some common substitutions used in integral evaluation include trigonometric substitutions, u-substitution, and partial fraction decomposition. Each substitution is chosen based on the structure of the integral and the goal of simplifying it.
Integral substitution is useful because it allows for the evaluation of more complex integrals that cannot be solved using basic integration rules. It also provides a way to simplify integrals and make them easier to solve. Additionally, it is a key technique in many applications of calculus, such as finding areas and volumes of irregular shapes.