7. Antiderivatives & Indefinite Integrals

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

4 sec 3x tan 3x

Checking Antiderivative Formulas

Verify the formulas in Exercises 57–62 by differentiation.

∫(3x + 5)⁻² dx = −(3x + 5)⁻¹/3 + C

Checking Antiderivative Formulas

Verify the formulas in Exercises 57–62 by differentiation.

∫csc²((x − 1)/3)dx = −3cot((x − 1)/3) + C

Checking Antiderivative Formulas

Right, or wrong? Say which for each formula and give a brief reason for each answer.

∫xsinx dx = -x cos x + C

Checking Antiderivative Formulas

Right, or wrong? Say which for each formula and give a brief reason for each answer.

∫tanθ sec²θ dθ = sec³θ / 3 + C

Checking Antiderivative Formulas

Right, or wrong? Say which for each formula and give a brief reason for each answer.

∫tanθ sec²θ dθ = (1/2) sec²θ + C

Checking Antiderivative Formulas

Right, or wrong? Say which for each formula and give a brief reason for each answer.

∫3(2x + 1)² dx = (2x + 1)³ + C

Checking Antiderivative Formulas

Right, or wrong? Say which for each formula and give a brief reason for each answer.

∫√(2x + 1) dx = √(x² + x +C)

Checking Antiderivative Formulas

Right, or wrong? Give a brief reason why.

∫−15(x + 3)² / (x − 2)⁴ dx = ((x + 3)/(x − 2))³ + C

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

−πsin πx