Integration

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Using the power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, where n ≠ -1. Here, n = 3. ∫ x³ dx = x⁴/4 + C.

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Standard integral: ∫ sin x dx = -cos x + C. Verification: d/dx(-cos x) = sin x.

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Standard integral: ∫ sec²x dx = tan x + C. Verification: d/dx(tan x) = sec²x.

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Standard integral: ∫ 1/x dx = log|x| + C. Note: Absolute value ensures domain for x < 0.

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Standard integral: ∫ eˣ dx = eˣ + C. Verification: d/dx(eˣ) = eˣ.

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Use linearity: ∫[f(x) + g(x)] dx = ∫f(x)dx + ∫g(x)dx. ∫x² dx + ∫3x dx + ∫2 dx = x³/3 + 3x²/2 + 2x + C.

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Constant multiple rule: ∫k f(x) dx = k ∫f(x) dx. 5 ∫cos x dx = 5 sin x + C.

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Use substitution: Let u = 2x + 3 → du = 2 dx → dx = du/2. ∫sin u (du/2) = (1/2)∫sin u du = (1/2)(-cos u) + C = -½ cos(2x+3) + C.