Calculus Practice Exams

Answers.


Problem 1. Evaluate

displaymath237

Answer. This is a definite integral. One way to handle it is to find an antiderivative of tex2html_wrap_inline247 and then evaluate the result at 0 and tex2html_wrap_inline249 .
Use integration by parts technique. Set

displaymath251

This gives

displaymath253

The technique of integrating rational functions gives

displaymath255

After easy calculations, we get

displaymath257

Now that we have the antiderivative we use it to get

displaymath259

Problem 2. Evaluate

displaymath238

Answer. We will use Substitution Techniques. Set

displaymath261

We have tex2html_wrap_inline263 . The new integral is

displaymath265

The technique of integrating rational functions gives

displaymath267

Since

displaymath269

we have

displaymath271

Do not forget to go back to the variable x, we have

displaymath275

Problem 3. Evaluate

displaymath239

Answer. Note that this integral is direct from the formula

displaymath277

Indeed, if we let tex2html_wrap_inline279 , then we have

displaymath281

in other words, we have

displaymath283

On the other hand, we may want to use the technique of integrating rational functions of tex2html_wrap_inline285 and tex2html_wrap_inline287 . Let us show the main points of this technique on this example for the sake of being somehow more complete!!!

Set tex2html_wrap_inline289 . Then we have

displaymath291

This gives

displaymath293

Easy algebraic computations yield:

displaymath295

Since

displaymath297

then we have

displaymath299

Back to the variable x, we have

displaymath303

Using the half-angle formula for the cosine we obtain

displaymath305

and thus finally

displaymath307

Problem 4. Evaluate

displaymath240

Answer. If we complete the square we get

displaymath309

which suggests the substitution tex2html_wrap_inline311 . Hence tex2html_wrap_inline313 . Therefore we have

displaymath315

Back to the variable x, we get

displaymath319

Problem 5. Evaluate

displaymath241

Answer. We have

displaymath321

We have

displaymath323

and

displaymath325

Putting the two together we get

displaymath327


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