Answer:
Explanation:
It seems like you've provided the first step for three different integrals, and you want to match each first step to the appropriate integral. Let's match them up:
Save a factor of sec^2(x) and use sec^2(x) = 1 + tan^2(x) to express the remaining factors in terms of tan(x).
This step is typically used when you're dealing with integrals involving secant and tangent functions. It helps simplify the integral by converting secant and tangent into a single variable, tan(x).
Perform u-substitution with u = x^2.
This step is often used when you have an integral where you can simplify the integrand by letting u equal a certain expression within the integral, making it easier to integrate.
Integration by parts, where you choose one function to differentiate and the other to integrate.
Integration by parts is a technique used to integrate products of two functions, typically when you have a product of functions and you want to break it down into more manageable parts.
Trigonometric substitution with a trigonometric identity like sin^2(x) + cos^2(x) = 1.
Trigonometric substitution is used when you have an integral that can be simplified by substituting trigonometric functions, typically using trigonometric identities.
The integral cannot be evaluated.
This is self-explanatory; if the integral is too complex or doesn't have an elementary antiderivative, it cannot be evaluated using standard methods.
Now, you can match each first step to the appropriate integral based on the technique required:
Integral of sec(x) dx:
The first step is to save a factor of sec^2(x) and use sec^2(x) = 1 + tan^2(x) to express the remaining factors in terms of tan(x).
Integral of x^2 e^(x^3) dx:
The first step is to perform u-substitution with u = x^2.
Integral of ln(x) dx:
The first step for this integral would be integration by parts, where you choose one function to differentiate and the other to integrate.
Integral of sqrt(1 - x^2) dx:
The first step is trigonometric substitution with a trigonometric identity like sin^2(x) + cos^2(x) = 1.
Integral of e^(x^2) dx:
The first step for this integral would be the standard notation for integration; there's no special technique mentioned here.
So, the first steps have been matched to the appropriate integrals.