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

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.

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 .

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

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: ${\int }_0^{\frac{\pi }{2}}3{\cos }^2\left(x\right) dx$.

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