7. Antiderivatives & Indefinite Integrals

Evaluate the integral. (Use c for the constant of integration.) $\int 4{\sin }^2\left(x\right){\cos }^3\left(x\right)dx$

Evaluate the integral: ${\int }_0^{\pi }x\sin \left(x\right)\cos \left(x\right)dx$

Evaluate the integral: ${\int }_{}^{}\arctan \left(\frac{1}{x}\right)dx$

5. What is a reduction formula?

9–61. Trigonometric integrals Evaluate the following integrals.

45. ∫ sec²x tan¹ᐟ²x dx

7–84. Evaluate the following integrals.

82. ∫ 1/(1 + tanx) dx

Use Table 5.6 to evaluate the following indefinite integrals.                                                                                                               

 (b) ∫ sec 5𝓍 tan 5𝓍 d𝓍

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.                                                                                  

 ∫ (sin⁵ 𝓍 + 3 sin³ 𝓍― sin 𝓍) cos 𝓍 d𝓍

Evaluate the integrals in Exercises 1–22.

∫ cos³(4x) dx

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

∫ x² sin(x³) dx

Evaluate the integrals in Exercises 33–52.

∫ cot⁶(2x) dx

7–84. Evaluate the following integrals.

49. ∫ tan³x · sec⁹x dx

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

29. ∫ cos⁴ x/sin⁶ x dx

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

$g\(\left\)(x\(\right\))=\(\int\]\left\)(\(\sin\)^2x-100\(\csc\) x\(\cot\) x+\(\cos\)^2x\(\right\))dx$

9–61. Trigonometric integrals Evaluate the following integrals.

32. ∫ cot⁵(3x) dx