7. Antiderivatives & Indefinite Integrals

Evaluate the indefinite integral: $\int 4x dx$

Evaluate the integral. (Use c for the constant of integration.) $\int x^2\ln \left(x\right) dx$

Evaluate the indefinite integral as an infinite series: ${\int }_{}^{}\frac{cos\left(x\right)-1}{x}dx$.

7–64. Integration review Evaluate the following integrals.

7. ∫ dx / (3 - 5x)^4

Evaluate the indefinite integral. (Use c for the constant of integration.)
$${\int }_{}^{}{\cos }^{-1}\left(x\right) dx$$

92–98. Evaluate the following integrals.

94. ∫ (dt / (t³ + 1))

Evaluate the indefinite integral: $\int 9x^2(a^2-x^2)dx$

Use Table 5.6 to evaluate the following indefinite integrals.                                                                                                               

 (e) ∫ d𝓍/(81 + 9𝓍²) (Hint: Factor a 9 out of the denominator first.)  

Evaluate the indefinite integral: ${\int }_{}^{}(x^3-8x^2-7x+10)dx$

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. ∫(3/(x² + 4)) dx = ∫(3/x²) dx + ∫(3/4) dx.

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.

∫ (5s + 3)² ds

Evaluate the following integrals.

∫ x/(x² + 6x + 18) dx

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

30. ∫ x³√(1 - x²) dx

Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.

∫ 1/(x(ln(x))²) dx

Evaluate the integral. (Use c for the constant of integration.) ${\int }_{}^{}{\sin }^{-1}\left(x\right)dx$