Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ, ƒ', and ƒ'' are continuous functions for all real numbers.                                                                                                                                                           
                                                                                                                                                                    
(c) ∫ sin 2𝓍 d𝓍 = 2 ∫ sin 𝓍 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

Integrals with sin² 𝓍 and cos² 𝓍 Evaluate the following integrals.                                                                                                             

 ∫ sin² 𝓍 d𝓍

76-81. Table of integrals Use a table of integrals to evaluate the following integrals.

79. ∫ sec⁵x dx

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

d. ∫2 sin x cos x dx = −(1/2) cos 2x + C.

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

$g\left(\theta\right)=\int^{}\left(5\sec^2\theta-2\csc^2\theta\right)d\theta$

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

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

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

Evaluate the integral: ${\int }_1^7{w}^2\ln \left(w\right) dw$