 Calculus Volume 2

# Review Exercises

Calculus Volume 2Review Exercises

### Review Exercises

For the following exercises, determine whether the statement is true or false. Justify your answer with a proof or a counterexample.

408 .

$∫exsin(x)dx∫exsin(x)dx$ cannot be integrated by parts.

409 .

$∫1x4+1dx∫1x4+1dx$ cannot be integrated using partial fractions.

410 .

In numerical integration, increasing the number of points decreases the error.

411 .

Integration by parts can always yield the integral.

For the following exercises, evaluate the integral using the specified method.

412 .

$∫x2sin(4x)dx∫x2sin(4x)dx$ using integration by parts

413 .

$∫1x2x2+16dx∫1x2x2+16dx$ using trigonometric substitution

414 .

$∫xln(x)dx∫xln(x)dx$ using integration by parts

415 .

$∫3xx3+2x2−5x−6dx∫3xx3+2x2−5x−6dx$ using partial fractions

416 .

$∫x5(4x2+4)5/2dx∫x5(4x2+4)5/2dx$ using trigonometric substitution

417 .

$∫4−sin2(x)sin2(x)cos(x)dx∫4−sin2(x)sin2(x)cos(x)dx$ using a table of integrals or a CAS

For the following exercises, integrate using whatever method you choose.

418 .

$∫ sin 2 ( x ) cos 2 ( x ) d x ∫ sin 2 ( x ) cos 2 ( x ) d x$

419 .

$∫ x 3 x 2 + 2 d x ∫ x 3 x 2 + 2 d x$

420 .

$∫ 3 x 2 + 1 x 4 − 2 x 3 − x 2 + 2 x d x ∫ 3 x 2 + 1 x 4 − 2 x 3 − x 2 + 2 x d x$

421 .

$∫ 1 x 4 + 4 d x ∫ 1 x 4 + 4 d x$

422 .

$∫ 3 + 16 x 4 x 4 d x ∫ 3 + 16 x 4 x 4 d x$

For the following exercises, approximate the integrals using the midpoint rule, trapezoidal rule, and Simpson’s rule using four subintervals, rounding to three decimals.

423 .

[T] $∫12x5+2dx∫12x5+2dx$

424 .

[T] $∫0πe−sin(x2)dx∫0πe−sin(x2)dx$

425 .

[T] $∫14ln(1/x)xdx∫14ln(1/x)xdx$

For the following exercises, evaluate the integrals, if possible.

426 .

$∫1∞1xndx,∫1∞1xndx,$ for what values of $nn$ does this integral converge or diverge?

427 .

$∫ 1 ∞ e − x x d x ∫ 1 ∞ e − x x d x$

For the following exercises, consider the gamma function given by $Γ(a)=∫0∞e−yya−1dy.Γ(a)=∫0∞e−yya−1dy.$

428 .

Show that $Γ(a)=(a−1)Γ(a−1).Γ(a)=(a−1)Γ(a−1).$

429 .

Extend to show that $Γ(a)=(a−1)!,Γ(a)=(a−1)!,$ assuming $aa$ is a positive integer.

The fastest car in the world, the Bugati Veyron, can reach a top speed of 408 km/h. The graph represents its velocity. 430 .

[T] Use the graph to estimate the velocity every 20 sec and fit to a graph of the form $v(t)=aexpbxsin(cx)+d.v(t)=aexpbxsin(cx)+d.$ (Hint: Consider the time units.)

431 .

[T] Using your function from the previous problem, find exactly how far the Bugati Veyron traveled in the 1 min 40 sec included in the graph.

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