Selecting an appropriate integration technique requires recognizing products of exponential and trigonometric functions. The integrand $e^{2x}\sin(2x)$ is a product where both functions have the same argument coefficient (2). This structure requires repeated integration by parts, typically applied twice to obtain a system that can be solved for the original integral. Substitution using $u = \sin(2x)$ wouldn't work effectively because the exponential function doesn't integrate to something involving sine. When you have products of exponential and trigonometric functions with matching coefficients, expect to use integration by parts repeatedly until a pattern emerges that allows solving for the original integral.

Selecting the appropriate technique for antidifferentiation is crucial for evaluating integrals efficiently. For the integral ∫ x $e^{3x}$ dx from 0 to 2, integration by parts is most appropriate because the integrand is a product of a polynomial and an exponential function. Set u = x and dv = $e^{3x}$ dx, so du = dx and v = (1/3) $e^{3x}$, leading to uv - ∫ v du, which simplifies easily. This method reduces the power of x, making the remaining integral straightforward. A tempting distractor like u-substitution might be considered if one mistakes the exponential for a composite function, but it fails because there's no clear inner function whose derivative matches the rest of the integrand. Recognize products of polynomials and exponentials or trig functions as prime candidates for integration by parts.

Selecting an appropriate integration technique requires identifying substitution patterns in radical expressions. The integrand $\frac{2x+1}{\sqrt{x^2+x}}$ suggests substitution because the numerator $2x+1$ is the derivative of the expression under the radical (since $\frac{d}{dx}[x^2+x] = 2x+1$). Using $u = x^2+x$ gives $du = (2x+1)dx$, transforming the integral to $\int u^{-1/2},du$. Partial fractions isn't applicable to radical expressions, and trigonometric substitution would be more complex. When the numerator is the derivative of the expression under a radical, substitution using that expression provides the most direct solution path.

Selecting an appropriate integration technique requires recognizing radical expressions with quadratic forms. The integrand $\sqrt{9+x^2}$ has the form $\sqrt{a^2+x^2}$ where $a = 3$. This structure specifically calls for trigonometric substitution using $x = 3\tan\theta$, which transforms the radical into $3\sec\theta$. Substitution using $u = 9+x^2$ wouldn't work because we don't have $2x$ as a factor in the integrand. Integration by parts isn't suitable since we don't have a clear product structure. When encountering $\sqrt{a^2+x^2}$, $\sqrt{a^2-x^2}$, or $\sqrt{x^2-a^2}$, trigonometric substitution is the standard technique for eliminating the radical.

Selecting the right integration technique requires identifying substitution opportunities. The integrand $\frac{4x}{(x^2+1)^3}$ contains $4x$ in the numerator, which is twice the derivative of $x^2+1$ (since $\frac{d}{dx}[x^2+1] = 2x$). Using substitution $u = x^2+1$ gives $du = 2x,dx$, so $4x,dx = 2du$. The integral becomes $2\int u^{-3},du$, which is straightforward. Trigonometric substitution would be much more complex and unnecessary given this clean substitution. When the numerator is a constant multiple of the derivative of an expression in the denominator, substitution using that expression is the most efficient approach.

Selecting the right integration technique involves recognizing improper rational functions. The integrand $\frac{x^3}{x^2+1}$ has a numerator degree greater than the denominator degree, requiring polynomial long division first. Dividing gives $\frac{x^3}{x^2+1} = x - \frac{x}{x^2+1}$, creating $\int x,dx - \int \frac{x}{x^2+1},dx$ where the second integral uses substitution. Integration by parts would be unnecessarily complex for this rational function structure. When dealing with improper rational functions (numerator degree ≥ denominator degree), always perform polynomial long division before applying other integration techniques.

Selecting an appropriate integration technique involves recognizing inverse trigonometric integration by parts. The integrand $\arctan x$ is an inverse trigonometric function alone, which requires integration by parts with $u = \arctan x$ and $dv = dx$. This gives $du = \frac{1}{1+x^2}dx$ and $v = x$, leading to $x\arctan x - \int \frac{x}{1+x^2}dx$. The remaining integral is manageable with substitution. Substitution using $u = \arctan x$ alone wouldn't work since we don't have $\frac{1}{1+x^2}$ as a factor. For integrals of inverse trigonometric functions by themselves, integration by parts is the standard approach.

Selecting an appropriate integration technique involves recognizing difference of squares patterns. The expression $\frac{1}{x^2-9}$ has a denominator that factors as $(x-3)(x+3)$, indicating partial fraction decomposition is needed. The partial fraction form is $\frac{A}{x-3} + \frac{B}{x+3}$, which leads to logarithmic terms after integration. Trigonometric substitution isn't the most direct approach here since the quadratic is factorable over the reals. When you have a rational function with a denominator that's a difference of squares (or any factorable quadratic), partial fractions typically provides a more straightforward path than trigonometric substitution.

Selecting an appropriate integration technique requires recognizing substitution with radicals involving square roots. The integrand $\frac{\sqrt{x}}{1+x}$ suggests substitution $u = \sqrt{x}$, which gives $x = u^2$ and $dx = 2u,du$. The integral transforms to $\int \frac{u}{1+u^2} \cdot 2u,du = 2\int \frac{u^2}{1+u^2},du$, which can be handled with polynomial division. Integration by parts would be complex due to the mixed radical and algebraic structure. When dealing with integrands involving square roots in both numerator and a function of the variable, substitution using the square root often simplifies the expression significantly.

Selecting an appropriate integration technique requires recognizing trigonometric substitution with radicals. The integrand $\sqrt{1+4x^2}$ can be rewritten as $\sqrt{1+(2x)^2}$, which has the form $\sqrt{a^2+u^2}$ where $a = 1$ and $u = 2x$. This suggests trigonometric substitution using $2x = \tan\theta$, which transforms the radical into $\sec\theta$. Partial fractions doesn't apply to radical expressions, and simple substitution wouldn't eliminate the radical effectively. When encountering $\sqrt{a^2+u^2}$ forms, trigonometric substitution with $u = a\tan\theta$ is the standard technique for handling the radical expression.