91影视

Prove the integration formula.

$鈭�\ln xdx=x\ln x-x+c$

(a) by applying integration by parts to $鈭�\ln xdx$.

(b) by differentiating$x\ln x-x$.

Use the chain rule to prove the formula for integration by substitution:

$鈭�f\text{'}\left(u\right(x\left)\right)u\text{'}\left(x\right)dx=f\left(u\right(x\left)\right)+C.$

We can extend the technique of trigonometric substitution to the hyperbolic functions. Use Theorem 2.20 and the identity ${\cosh }^{2}x-{\sinh }^{2}x=1$to solve each integral in Exercises 87鈥� 90 with an appropriate hyperbolic substitution $x=a\sinh u\mathrm{or}x=a\cos hu$. (These exercises involve hyperbolic functions.)

$鈭�\frac{x^2}{{\left(x^2+1\right)}^{3/2}}dx$

Describe strategy for solving the type of integral given.

$鈭�{\cos }^{k}xdx,k$ is even.

Why does it make sense that ${鈭�}_0^1\frac{1}{x^p}dx$diverges when $p鈮�1$? Consider how $\frac{1}{x^p}$compares with $\frac{1}{x}$in this case.

If $f$is concave down on all of on [a, b], which of the given approximations is guaranteed to be an over-approximation for ${鈭�}_a^{b}f\left(x\right)dx$? (Select all that apply.)

(a) left sum

(b) right sum

(c) trapezoid sum

(d) midpoint sum

Why don鈥檛 we need to have a square root involved in order to apply trigonometric substitution with the tangent? In other words, why can we use the substitution $x=a\tan 鈦�u$when we see $x^2+a^2$, even though we can鈥檛 use the substitution $x=a\sin 鈦�u$unless the integrand involves the square root of$a^2鈭�x^2$? (Hint: Think about domains.)

For each function u(x) in Exercises 9鈥�12, write the differential du in terms of the differential dx.

$u\left(x\right)=x^2+x+1$

In Exercises 9鈥�11, suppose that you apply polynomial long division to divide a polynomial p(x) by a polynomial q(x) with the goal of obtaining an expression of the form $\frac{p\left(x\right)}{q\left(x\right)}=m\left(x\right)+\frac{R\left(x\right)}{q\left(x\right)}$

If the degree of p(x) is two greater than the degree of q(x), what can you say about the degrees of m(x) andR(x)?

Use whatever method you like to solve each of the given definite and indefinite integrals. These integrals are neither in order of difficulty nor in order of technique. Many of the integrals can be solved in more than one way.

$鈭�2蟺x\left(8-x^{3/2}\right)dx$