Selecting the appropriate technique for antidifferentiation is a key skill in calculus, as it involves recognizing the structure of the integrand to apply the most efficient method. For the integral ∫ 1/(x (ln x)²) dx, substitution with u = ln x is most appropriate because du = (1/x) dx directly matches the 1/x factor, simplifying the integral to ∫ 1/u² du. This substitution transforms the composite function into a basic power rule integral. The presence of ln x and its derivative 1/x in the integrand makes this a clear candidate for logarithmic substitution. While integration by parts might be tempting for products, it fails here as it would complicate the integral without simplifying the logarithmic term. Always look for inner functions whose derivatives appear in the integrand to identify substitution opportunities.

This question tests the skill of selecting an appropriate technique for antidifferentiation. The integrand $\sin x / (1 + \cos x)$ has $\sin x$ in the numerator, which is the negative derivative of $\cos x$ in the denominator, suggesting substitution. Letting $u = 1 + \cos x$, $du = -\sin x , dx$, gives $-\int \frac{du}{u} = -\ln|u| + C$. This simplifies the trig rational directly. Partial fractions might be tempting if misreading as rational in x, but the trig functions make it inapplicable. Identify substitutions in trig rationals where the numerator matches the derivative of part of the denominator.

Selecting the appropriate technique for antidifferentiation is a key skill in calculus, as it involves recognizing the structure of the integrand to apply the most efficient method. For the integral ∫ (1/x - 1/x²) dx, splitting into simpler terms and using basic rules is most appropriate because it becomes ∫ 1/x dx - ∫ $x^{-2}$ dx, yielding ln|x| + 1/x + C. Each term is a standard integral. No advanced techniques are needed for this algebraic separation. While substitution with u = 1/x might be tempting, it fails as it complicates the integral without necessity. Always simplify expressions by splitting before applying integration rules.

This question tests the skill of selecting an appropriate technique for antidifferentiation. The integrand features an exponential with a composite exponent and a polynomial factor that matches the derivative of that exponent, ideal for substitution. Setting u = x² - x gives du = (2x - 1) dx, transforming the integral into ∫ $e^u$ du, which is simply $e^u$ + C. This substitution captures the entire structure efficiently without needing decomposition or parts. Integration by parts might tempt due to the product form, but it would complicate things unnecessarily since the substitution directly simplifies the exponential composition. To spot substitution, check if a factor in the integrand is the derivative of the inner function of a composite.

This question tests the skill of selecting an appropriate technique for antidifferentiation. The integrand $\frac{e^{2x}}{1 + e^{2x}}$ resembles a form where the numerator is related to the derivative of the denominator, suggesting substitution. Letting $u = 1 + e^{2x}$, $du = 2 e^{2x} , dx$, gives $\frac{1}{2} \int \frac{du}{u} = \frac{1}{2} \ln|u| + C$. This substitution simplifies the exponential rational function directly. Integration by parts might tempt for the apparent product, but there's no clear product, making it inefficient compared to sub. Spot substitution in rational functions with exponentials when the numerator matches part of the denominator's derivative.

This question tests the skill of selecting an appropriate technique for antidifferentiation. The integrand is a rational function where the numerator x² is nearly the derivative of the denominator x³ + 7, which is 3x², making substitution ideal. Setting u = x³ + 7 gives du = 3x² dx, so the integral is (1/3) ∫ du/u = (1/3) ln|u| + C. This matches perfectly, simplifying the polynomial ratio. Partial fractions might tempt for rational functions, but it fails here as the denominator isn't factored into linears or quadratics suitable for decomposition. Check if the numerator is a scalar multiple of the denominator's derivative to prioritize substitution over other rational techniques.

This question tests the skill of selecting an appropriate technique for antidifferentiation. The integrand x / √(x² - 16) has x in the numerator, half the derivative of x² - 16, making substitution appropriate. Setting u = x² - 16, du = 2x dx, gives (1/2) ∫ du / √u = √u + C. This handles the radical quadratic directly. Trigonometric substitution might be tempting for √(x² - a²), but sub is simpler here without introducing angles. Prioritize substitution for radicals when a linear factor matches the inner derivative.

Selecting the appropriate technique for antidifferentiation is a key skill in calculus, as it involves recognizing the structure of the integrand to apply the most efficient method. For the integral $\int \frac{5}{2x - 3} , dx$, the basic logarithm rule with linear substitution is most appropriate because it is a constant over a linear function, which integrates to $\frac{5}{2} \ln|2x - 3| + C$. A simple $u = 2x - 3$ substitution confirms $du = 2 , dx$, adjusting the constant accordingly. This form directly matches the integral of $\frac{1}{u} , du$ up to scalars. While partial fractions might be tempting for rational functions, it fails here as there's no need to decompose a single linear denominator. Always check if a rational integrand is already in its simplest form before considering decomposition.

Selecting the appropriate technique for antidifferentiation is a key skill in calculus, as it involves recognizing the structure of the integrand to apply the most efficient method. For the integral $\int \frac{2}{1 + x^2} , dx$, recognizing an inverse trigonometric derivative pattern is most appropriate because it matches the derivative of $\arctan(x)$, yielding $2 \arctan(x) + C$. The form $\frac{1}{1 + x^2}$ is the standard pattern for arctan. Adjusting the constant 2 makes it straightforward. While trigonometric substitution might be tempting, it fails as it's unnecessary and more complex for this basic form. Always identify if the integrand resembles known derivatives of inverse trig functions for quick solutions.

This question tests the skill of selecting an appropriate technique for antidifferentiation. The integrand is a product of sine and cosine with the same argument, which suggests using a trigonometric identity to simplify before integrating. The double-angle identity sin(2θ) = 2 sin θ cos θ allows rewriting sin(3x) cos(3x) as (1/2) sin(6x), leading to a straightforward integral of (1/2) ∫ sin(6x) dx. This algebraic manipulation fits perfectly for products of like trig functions. Integration by parts might be tempting for the product form, but it would lead to more complex terms without simplification, whereas the identity reduces it efficiently. Look for trig products that match known identities to simplify the integrand before choosing other methods.