7. Antiderivatives & Indefinite Integrals

7–64. Integration review Evaluate the following integrals.

38. ∫ x / (x⁴ + 2x² + 1) dx

68. Different methods

b. Evaluate ∫(cot x csc² x) dx using the substitution u=cscx.

64. Using a computer algebra system, it was determined that

∫x(x+1)^8 dx = (x^10)/10 + (8x^9)/9 + (7x^8)/2 + 8x^7 + (35x^6)/3 + (56x^5)/5 + 7x^4 + (8x^3)/3 + x^2/2 + C.

Use integration by substitution to evaluate ∫x(x+1)^8 dx.

Finding Indefinite Integrals

In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫(t√t + √t) / t² dt

Finding Indefinite Integrals

In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫(1 + cos 4t)/2 dt

7–64. Integration review Evaluate the following integrals.

28. ∫ (3x + 1) / √(4 - x²) dx

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

∫ x / (1 + √x) dx

Evaluating integrals Evaluate the following integrals.                                                                                                                                         

 ∫ d𝓍/[(tan⁻¹ 𝓍) (1 + 𝓍²)]

Finding Indefinite Integrals

Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫ sec θ/3 tan θ/3 dθ