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/√(4 - x²) dx, recognizing an inverse trigonometric derivative pattern is most appropriate because the form matches the derivative of arcsin(x/2), which is 1/√(4 - x²) after adjusting constants. This direct recognition allows for immediate antidifferentiation without additional steps. The integrand's square root of a quadratic in the denominator with a negative x² term signals this inverse sine pattern. While substitution with u = 4 - x² might seem tempting, it fails because the du = -2x dx introduces an x in the numerator that isn't present, requiring further adjustments. Always scan for standard integral forms like inverse trig functions when dealing with square roots of quadratics.

This question tests the skill of selecting an appropriate technique for antidifferentiation. The integrand is a rational function where the numerator is a constant multiple of the derivative of the denominator, making substitution a natural fit. By setting u = x² + 4, du = 2x dx, so the integral becomes 3 ∫ du/u, which simplifies to 3 ln|u| + C. This direct match between the numerator and the derivative of the inner function allows for straightforward integration. A tempting distractor like trigonometric substitution might seem appealing due to the quadratic in the denominator, but it is unnecessary here as the form doesn't require converting to a trig identity for simplification. Always look for cases where the numerator resembles the derivative of the denominator to recognize substitution opportunities.

This question tests the skill of selecting an appropriate technique for antidifferentiation. The integrand 1/(x+3)² is already in the form $(x+3)^{-2}$, which integrates directly via the power rule to -1/(x+3) + C. This basic rule applies perfectly to the negative exponent. No further manipulation is needed. Integration by parts might be considered unnecessarily for what looks like a product of 1 and 1/(x+3)², but it complicates a simple power integration. Rewrite rational expressions as powers to apply the basic antiderivative rules efficiently.

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 ∫ 4/√(2x + 1) dx from 1 to 4, substitution with u = 2x + 1 is most appropriate because du = 2 dx, so 2 ∫ $u^{-1/2}$ du, yielding 4 √u + C. This simplifies the square root in the denominator. The linear inside the root with a matching differential makes it ideal. While trigonometric substitution might be tempting for roots, it fails as it's overkill for this simple form. Always use substitution for compositions where the inner function's derivative is present.

This question tests the skill of selecting an appropriate technique for antidifferentiation. The integrand is ln x, a transcendental function without an elementary substitution or direct rule, making integration by parts the best choice. By setting u = ln x and dv = dx, it becomes ∫ u dv = uv - ∫ v du, yielding x ln x - ∫ x (1/x) dx = x ln x - x + C. This method effectively handles the logarithmic function multiplied implicitly by 1. Substitution with u = ln x might seem promising, but it fails because du = (1/x) dx, leaving no x to pair with, resulting in an incomplete transformation. Recognize integration by parts when the integrand is a product where one part is easy to integrate and the other to differentiate.

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 $\sec^2(5x)$ is the derivative of $\tan(5x)$, so a basic rule with a simple chain rule substitution applies. Letting $u = 5x$, $du = 5 , dx$, yields $\frac{1}{5} \int \sec^2 u , du = \frac{1}{5} \tan u + C$. This direct method leverages known trig antiderivatives. Integration by parts might be considered for trig functions, but it's overkill here as the basic rule suffices without recursion. Remember to use simple substitutions for composite trig functions that match standard antiderivative forms.

This question tests the skill of selecting an appropriate technique for antidifferentiation. The integrand is a product of x and arctan x, where neither has a simple antiderivative alone, making integration by parts suitable. Setting u = arctan x, dv = x dx leads to uv - ∫ v du = (x²/2) arctan x - ∫ (x²/2)/(1 + x²) dx, which can be resolved. This method handles the inverse trig and polynomial product well. Trigonometric substitution might seem appealing due to arctan, but it doesn't simplify the product form directly. Identify products of functions where one differentiates nicely and the other integrates easily for integration by parts.

This question tests the skill of selecting an appropriate technique for antidifferentiation. The integrand 1/(x² - 4) is a rational function with difference of squares denominator, factorable into (x-2)(x+2), perfect for partial fractions. Decomposing yields (1/4) [ln|x-2| - ln|x+2|] + C. This method handles the hyperbolic-like form efficiently. Trigonometric substitution might tempt due to the x² - a² structure, but partial fractions are simpler for this rational without radicals. For rational functions with linear factors, use partial fractions when direct integration isn't obvious.

This question tests the skill of selecting an appropriate technique for antidifferentiation. The integrand ln(x²)/x can be manipulated using log properties: ln(x²) = 2 ln x, so it becomes 2 ∫ (ln x)/x dx. Then substitution u = ln x, du = (1/x) dx, yields 2 ∫ u du = u² + C = (ln x)² + C. This combines algebra with sub effectively. Integration by parts might tempt for the apparent product, but the log property simplification makes sub more direct. Use log properties to simplify arguments before substituting when integrating logarithmic forms.