The skill here is integration using u-substitution, which simplifies the integral by replacing a composite function with a single variable. Recognize that the inner function is 3x - 2, whose derivative 3 requires adjusting the 6 to 2. Set u = 3x - 2, so du = 3 dx and the integral becomes 2 ∫ du/u. Integrating gives 2 ln|u| + C, or 2 ln|3x - 2| + C. A tempting distractor like 6 ln|3x - 2| + C fails because it doesn't divide by the 3 from du, resulting in too large a coefficient. To recognize opportunities for u-substitution, look for an inner function whose derivative is a constant multiple of the remaining factors in the integrand.

This integral requires the skill of integration by substitution, often called u-substitution. The numerator (2x - 1) matches the derivative of the inner quadratic x² - x + 6 in the denominator raised to the third power. Setting u = x² - x + 6 gives du = (2x - 1) dx, simplifying to ∫ $u^{-3}$ du. Integration results in -1/(2u²) + C, or -1/(2(x² - x + 6)²) + C. A tempting distractor is choice C, which omits the factor of 1/2, possibly from forgetting to divide by -2 when integrating $u^{-3}$. To recognize opportunities for u-substitution, look for a composite function where the derivative of the inner part appears as a factor in the integrand.

The skill here is integration using u-substitution, which simplifies the integral by replacing a composite function with a single variable. Recognize that the inner function is x² + 3x + 1, whose derivative 2x + 3 exactly matches the numerator. Set u = x² + 3x + 1, so du = (2x + 3) dx, transforming the integral into ∫ du / √u. Integrating gives 2 √u + C, or 2 √(x² + 3x + 1) + C. A tempting distractor like √(x² + 3x + 1) + C fails because it misses the factor of 2 from integrating $u^{-1/2}$. To recognize opportunities for u-substitution, look for an inner function whose derivative is a constant multiple of the remaining factors in the integrand.

The skill here is integrating using u-substitution. The denominator has a square root of a cubic, and the numerator 3x² + 1 is its derivative, so u = x³ + x + 5 with du = (3x² + 1) dx. This turns the integral into ∫ du / √u = 2√u + C. Back-substituting gives 2√(x³ + x + 5) + C. Choice C halves incorrectly, forgetting the 2 from integrating $u^{-1/2}$. Spot u as the inside of the root when the numerator is its derivative for these forms.

The skill here is integrating using u-substitution. The integrand has (4x³ - 3x² + $8)^2$ multiplied by its derivative 12x² - 6x. Set u = 4x³ - 3x² + 8 with du = (12x² - 6x) dx, yielding ∫ $u^2$ du = $u^3$ / 3 + C. This is (4x³ - 3x² + $8)^3$ / 3 + C. Choice C divides by 12 incorrectly, mishandling the power rule. Look for polynomials raised to powers multiplied by their derivatives for power-rule substitutions.

The skill here is integration using u-substitution, which simplifies the integral by replacing a composite function with a single variable. Recognize that the inner function is 7x² + 3, whose derivative 14x exactly matches the numerator. Set u = 7x² + 3, so du = 14x dx, transforming the integral into ∫ du / √u. Integrating gives 2 √u + C, or 2 √(7x² + 3) + C. A tempting distractor like √(7x² + 3) + C fails because it misses the 2 from integrating $u^{-1/2}$. To recognize opportunities for u-substitution, look for an inner function whose derivative is a constant multiple of the remaining factors in the integrand.

This integral requires the skill of integration by substitution, often called u-substitution. The argument $7x - 3$ is the inner function of cosine, with derivative $7$. Setting $u = 7x - 3$ gives $du = 7 , dx$, so $dx = \frac{du}{7}$, and the integral is $\frac{1}{7} \int \cos u , du$. This integrates to $\frac{1}{7} \sin u + C$, or $\frac{1}{7} \sin(7x - 3) + C$. A tempting distractor is choice C, which uses cosine instead of sine, possibly from confusing the integral of cosine. To recognize opportunities for u-substitution, look for a composite function where the derivative of the inner part appears as a factor in the integrand.

This integral requires the skill of integration by substitution, often called u-substitution. The numerator 6x^5 is the derivative of x^6 + 2 in the denominator raised to 4. Let $u = x^6 + 2$, $du = 6x^5 , dx$, simplifying to $\int u^{-4} , du$. Integration yields $-\frac{1}{3u^3} + C$, or $-\frac{1}{3(x^6 + 2)^3} + C$. A tempting distractor is choice A, omitting the 1/3, perhaps from forgetting to divide by -3 in the power rule. To recognize opportunities for u-substitution, look for a composite function where the derivative of the inner part appears as a factor in the integrand.

The skill here is integrating using u-substitution. The integrand is a constant times a power of a linear function, so set u = 2x + 1 with du = 2 dx. Rewrite the integral as (9/2) ∫ $u^8$ du, which integrates to $(9/2)(u^9$/9) = $(1/2)u^9$ + C. This simplifies to (2x + $1)^9$ / 2 + C. Choice D divides by an extra factor, likely from mishandling the chain rule constant. Identify compositions where the outer function is a power and the inner's derivative appears as a factor.

The skill here is integrating using u-substitution. The denominator is x² + 1, with numerator 2x as its derivative. Set u = x² + 1 with du = 2x dx, yielding ∫ du/u = ln|u| + C. This is ln(x² + 1) + C. Choice B halves incorrectly, as the 2 is in du. Look for non-linear denominators with derivative numerators for logarithmic results.