Use integration formulas to solve each integral. You may...

Use integration formulas to solve each integral. You may have to use algebra, educated guessand-check, and/or recognize an integrand as the result of a product, quotient, or chain nule calculation. Check each of your answers by differentiating. (Hint for Exercise 54 : $\left.\tan x=\frac{\sin x}{\cos x}\right)$ $$ \int\left(x^{3}+4\right)^{2} d x $$

Use integration formulas to solve each integral. You may have to use algebra, educated guessand-check, and/or recognize an integrand as the result of a product, quotient, or chain nule calculation. Check each of your answers by differentiating. (Hint for Exercise 54 :
$\left.\tan x=\frac{\sin x}{\cos x}\right)$
$$
\int\left(x^{3}+4\right)^{2} d x
$$

Key Concepts

Integration Formulas

Integration formulas are standard antiderivative results for basic functions (such as power functions, exponential functions, trigonometric functions, etc.) that form the building blocks for solving more complex integrals. Mastering these formulas allows one to recognize patterns within an integrand and apply the appropriate formula without reinventing the underlying process each time.

Substitution Method

The substitution method (or u?substitution) is a key technique in integration that simplifies the integrand by changing variables to reduce a composite function into a more manageable form. It involves identifying a part of the integrand whose derivative also appears in the expression and then substituting it with a new variable. This method essentially reverses the chain rule used in differentiation.

Expansion of Polynomials

Expanding polynomials is an algebraic technique where products or powers of binomials are multiplied out to represent the expression as a sum of individual terms. This approach can simplify integration, as it allows one to apply the power rule of integration separately to each term. In some cases, expanding the integrand is more straightforward than attempting a substitution.

Differentiation for Verification

Differentiation for verification is a method used to check the correctness of an antiderivative by differentiating the result and ensuring it matches the original integrand. This cross-check utilizes the fundamental relationship between differentiation and integration and solidifies confidence in the solution obtained through integration techniques.

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