Deragz

2022-01-01

I need help calculating two integrals

1)${\int}_{1}^{2}\sqrt{4+\frac{1}{x}}dx$

2)${\int}_{0}^{\frac{2}{\pi}}{x}^{n}\mathrm{sin}\left(x\right)dx$

1)

2)

lovagwb

Beginner2022-01-02Added 50 answers

For the first problem:

$\int \sqrt{4+\frac{1}{x}}dx=\int \frac{\sqrt{4x+1}}{\sqrt{x}}dx$

Let$u=\sqrt{x},\text{}x={u}^{2},\text{}2udu=dx$

$\int \frac{\sqrt{4{u}^{2}+1}}{u}2udu$

Try some trigonometric substitution.

Let

Try some trigonometric substitution.

Lindsey Gamble

Beginner2022-01-03Added 38 answers

For the second problem:

Set

$I\left(n\right)=\int {x}^{n}\mathrm{sin}\left(x\right)dx$

Let$u={x}^{n},\text{}dv=\mathrm{sin}x,\text{}du=n{x}^{n-1},\text{}v=-\mathrm{cos}x$ by integration be parts.

$I\left(n\right)={x}^{n}\times \mathrm{cos}x+n\int {x}^{n-1}\mathrm{cos}xdx$

Let$u={x}^{n},\text{}dv=\mathrm{cos}x,\text{}du=(n-1){x}^{n-2},\text{}v=\mathrm{sin}x$ by integration be parts.

$I\left(n\right)={x}^{n}\times \mathrm{cos}x+n({x}^{n-1}\mathrm{sin}x-(n-1)\int {x}^{n-2}\mathrm{sin}xdx)={x}^{n}\times \mathrm{cos}x+n{x}^{n-1}\mathrm{sin}x-n(n-1)I(n-2)$

Then use mathematic induction you will get the formula.

Set

Let

Let

Then use mathematic induction you will get the formula.

karton

Expert2022-01-09Added 613 answers

You can indeed work out the second integral by parts.

Repeat with the cosine integral,

This gives you the recurrence relations

So that

When you decrease n, you will eventually reach n=1 or n=0, you need to explictly compute

then

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function

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