Indefinite Integration — The Power Rule
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Question
What does the ∫ symbol mean?
Answer
"Integrate with respect to x." The integral symbol ∫ paired with dx means find the antiderivative — the reverse of differentiation.
💡 Hint
It is the elongated S for "sum".
Question
State the power rule for integration.
Answer
∫xⁿ dx = xⁿ⁺¹/(n+1) + C, provided n ≠ −1. Add 1 to the power, divide by the new power, add C.
💡 Hint
Opposite of the power rule for differentiation.
Question
Why must you always include +C in an indefinite integral?
Answer
Because constants disappear when you differentiate. Infinitely many functions have the same derivative — +C represents all of them.
💡 Hint
Example: d/dx(x²+5) = d/dx(x²−7) = 2x.
Question
∫(4x³ − 6x + 2) dx = ?
Answer
x⁴ − 3x² + 2x + C. Integrate each term: 4·x⁴/4 = x⁴, 6·x²/2 = 3x², 2·x = 2x.
💡 Hint
Integrate term by term.
Question
What is the first step when integrating a product like x(x+3)?
Answer
Expand the brackets first: x(x+3) = x² + 3x. Then integrate: x³/3 + 3x²/2 + C.
💡 Hint
You cannot integrate products directly — expand first.
Question
∫x^(1/2) dx = ?
Answer
(2/3)x^(3/2) + C. Add 1: 1/2 + 1 = 3/2. Divide by 3/2: divide by 3/2 = multiply by 2/3.
💡 Hint
Don't panic with fractions — same rule applies.
Question
How do you check an integral is correct?
Answer
Differentiate your answer. If you get back the original integrand, your integral is correct.
💡 Hint
Differentiation and integration are inverse operations.
Question
∫(x² − 3)/x dx = ?
Answer
Rewrite: x²/x − 3/x = x − 3x⁻¹. Integrate: x²/2 − 3ln|x| + C.
💡 Hint
Split the fraction first, then use power rule.
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