Antiderivatives

Learn antiderivative power rule, the reciprocal exception.

Lesson 1 of 32 parts Practice Flashcards

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Antiderivative Power Rule

Objective: Explain antiderivative power rule, apply the stated method, and check its conditions.

For n≠−1 on a suitable interval, ∫xⁿdx=xⁿ⁺¹/(n+1)+C. A constant coefficient carries through. The family differs by additive constants because differentiation removes them; differentiate a proposed antiderivative to check it. The exponent −1 is an exception to the power shortcut.

Formulas

$∫axⁿdx=axⁿ⁺¹/(n+1)+C, n≠−1$

Worked examples

Try itAn antiderivative of 42x² is Kx³+C. Find K.Show answer

Step 1: Increase the exponent to 3 and divide by 3. Step 2: F(x)=(42/3)x³+C. Step 3: The coefficient K is 14; differentiation checks it.

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Antiderivative Power Rule

Objective: Explain antiderivative power rule, apply the stated method, and check its conditions.

Formulas

$∫axⁿdx=axⁿ⁺¹/(n+1)+C, n≠−1$

Worked examples

Try itAn antiderivative of 42x² is Kx³+C. Find K.Show answer

Step 1: Increase the exponent to 3 and divide by 3. Step 2: F(x)=(42/3)x³+C. Step 3: The coefficient K is 14; differentiation checks it.

The Reciprocal Exception

Objective: Explain the reciprocal exception, apply the stated method, and check its conditions.

On intervals excluding zero, an antiderivative of 1/x is ln|x|. On x>0 this is ln x. The power-rule antiderivative formula cannot use n=−1 because it would divide by zero. Preserve the interval and include an arbitrary constant. ln|x| handles negative intervals too; the function remains undefined at zero.

Formulas

$∫(1/x)dx=ln|x|+C$

Worked examples

Try itFor x>0, an antiderivative of 6/x is K ln x+C. Find K for the function with value f(7)=6/7.Show answer

Step 1: The derivative of ln x is 1/x on x>0. Step 2: Keep the coefficient 6. Step 3: K=6; the antiderivative is 6 ln x+C.

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