Antidifferentiation — Quiz 5

Questions in this quiz (10)

1. Evaluate $$\int \frac{6x}{\sqrt{16-x^4}}\,dx$$.

   • $$\sin^{-1}\left(\frac{x^2}{4}\right)+C$$
   • $$3\sin^{-1}\left(\frac{x^2}{4}\right)+C$$
   • $$\tan^{-1}\left(\frac{x^2}{4}\right)+C$$
   • $$\ln|16-x^4|+C$$

2. Evaluate $$\int \frac{3x^2}{x^3\sqrt{x^8-9}}\,dx$$.

   • $$\frac13\sec^{-1}(x^4)+C$$
   • $$\frac13\sec^{-1}(x^4)+C$$
   • $$\ln|x^8-9|+C$$
   • $$\frac34\sec^{-1}\left(\frac{x^4}{3}\right)+C$$

3. Evaluate $$\int \frac{\cos x}{\sin^2 x+4}\,dx$$.

   • $$\frac12\tan^{-1}\left(\frac{\sin x}{2}\right)+C$$
   • $$\ln|\sin x+4|+C$$
   • $$\sec x+C$$
   • $$-\tan x+C$$

4. Evaluate $$\int \frac{5\,dx}{\sqrt{25-25x^2}}$$.

   • $$\cos^{-1}(x)+C$$
   • $$\tan^{-1}(x)+C$$
   • $$\sin^{-1}(x)+C$$
   • $$\ln|x|+C$$

5. Evaluate $$\int \frac{\sec x}{\tan^2 x\sqrt{\sec^2 x-9}}\,dx$$.

   • $$\frac13\sec^{-1}\left(\frac{\sec x}{3}\right)+C$$
   • $$\frac13\sec^{-1}\left(\frac{\sec x}{3}\right)+C$$
   • $$\tan^{-1}(\sec x)+C$$
   • $$\sin x+C$$

6. Evaluate $$\int_1^4 x^{1/2}\left(3x^2-\frac{2}{\sqrt{x}}\right)\,dx$$.

   • $$38$$
   • $$42$$
   • $$46$$
   • $$\frac{720}{7}$$

7. Evaluate $$\int \frac{e^{\tan x}}{1+\tan^2 x}\,dx$$.

   • $$e^{\tan x}+C$$
   • $$\ln|e^{\tan x}|+C$$
   • $$\sec x+C$$
   • $$\tan x+C$$

8. If $$\int_0^4 f(x)\,dx=12$$ and $$\int_4^0 g(x)\,dx=8$$, evaluate $$\int_0^4 (2g(x)-f(x))\,dx$$.

   • $$4$$
   • $$8$$
   • $$-28$$
   • $$-8$$

9. Evaluate $$\int \frac{\cos\left(\frac{1}{x^3}\right)}{x^4}\,dx$$.

   • $$\sin\left(\frac{1}{x^3}\right)+C$$
   • $$-\frac13\sin\left(\frac{1}{x^3}\right)+C$$
   • $$\cos\left(\frac{1}{x^3}\right)+C$$
   • $$\ln|x|+C$$

10. Find the average value of $$f(x)=2^{\tan x}\sec x\tan x$$ on $$\left[0,\frac{\pi}{4}\right]$$.

    • $$2^{\tan x}\Big|_0^{\pi/4}$$
    • $$\frac{4-2}{\pi/4}$$
    • $$\frac{2}{\pi}(2-1)$$
    • $$0.83486224$$