7. Antiderivatives & Indefinite Integrals

For Exercises 49–52, complete the square before using an appropriate trigonometric substitution.

∫ 1 / √(x² - 2x + 5) 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.

123. ∫ √x * √(1 + √x) dx

Find $h\(\left\)(x\(\right\))$ by evaluating the following indefinite integral.

$h\(\left\)(x\(\right\))=x^{100}dx$

Variations on the substitution method Evaluate the following integrals.                                                                                                        

 ∫ y²/(y + 1)⁴ dy

Which of the following expressions is equivalent to the indefinite integral ${\int }_{}^{}(5x^3+11x+2)dx$?

Find the general indefinite integral. (Use c for the constant of integration.) $8x^{9}dx$

Find $h\left(x\right)$ by evaluating the following indefinite integral.

$h\left(x\right)=\int x^{8}dx$

37–56. Integrals Evaluate each integral.

∫ dx/x√(16 + x²)

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

3. ∫ (3x)/√(x + 4) dx

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.

18. ∫ dx / (225 − 16x²)

Use the given substitution to evaluate the following indefinite integrals. Check your answer by differentiating.                                                                                              

 ∫ (6𝓍 + 1) √(3𝓍² + 𝓍) d𝓍 , u = 3𝓍² + 𝓍

90–103. Indefinite integrals Determine the following indefinite integrals.

∫ (2x +1)² dx

Evaluate the indefinite integral: $\int 9x^2 dx$

Evaluate the integrals in Exercises 29–32 (b) using a trigonometric substitution.

∫ [x / √(4 − x²)] dx

Evaluating integrals Evaluate the following integrals.                                                                                                                                         

 ∫ (9𝓍⁸―7𝓍⁶) d𝓍