elbluffz1

2022-01-25

How is the integral $\frac{2}{\pi}{\int}_{0}^{\pi}{x}^{2}\mathrm{cos}\left(nx\right)dx=\frac{4{(-1)}^{n}}{{n}^{2}}$ ?

I thought it would be this :

$\frac{2}{\pi}{\int}_{0}^{\pi}{x}^{2}\mathrm{cos}\left(nx\right)dx=\frac{2}{\pi}{\int}_{0}^{\pi}{x}^{2}{(-1)}^{n}=\frac{2}{\pi}{(-1)}^{n}{\int}_{0}^{\pi}{x}^{2}=\frac{2}{\pi {(-1)}^{n}}{\left[\frac{{x}^{3}}{3}\right]}_{0}^{\pi}=\frac{2{(-1)}^{n}}{3{\pi}^{3}}$

But it is actually

$\frac{2}{\pi}{\int}_{0}^{\pi}{x}^{2}\mathrm{cos}\left(nx\right)dx=\frac{4{(-1)}^{n}}{{n}^{2}}$

I thought it would be this :

But it is actually

Georgia Ingram

Beginner2022-01-26Added 11 answers

You cannot say that $\mathrm{cos}nx={(-1)}^{n}$ . This is valid for $x=\pi$ . A counterexample would be $\mathrm{cos}n\cdot 0=1$ , and this doesnt

Eliza Norris

Beginner2022-01-27Added 15 answers

Hint: Two times integrating by parts we get

$\int}_{0}^{\pi}{x}^{2}\mathrm{cos}\left(nx\right)dx=\frac{{\pi}^{2}\mathrm{sin}\left(\pi n\right){n}^{2}+2n\mathrm{cos}\left(\pi n\right)\pi -2\mathrm{sin}\left(\pi n\right)}{{n}^{3}$

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

$f(x,y)={x}^{3}-6xy+8{y}^{3}$ $\frac{1}{\mathrm{sec}(x)}$ in derivative?

What is the derivative of $\mathrm{ln}(x+1)$?

What is the limit of $e}^{-x$ as $x\to \infty$?

What is the derivative of $f\left(x\right)={5}^{\mathrm{ln}x}$?

What is the derivative of $e}^{-2x$?

How to find $lim\frac{{e}^{t}-1}{t}$ as $t\to 0$ using l'Hospital's Rule?

What is the integral of $\sqrt{9-{x}^{2}}$?

What is the derivative of $f\left(x\right)=\mathrm{ln}\left[{x}^{9}{(x+3)}^{6}{({x}^{2}+7)}^{5}\right]$ ?

What Is the common difference or common ratio of the sequence 2, 5, 8, 11...?

How to find the derivative of $y={e}^{5x}$?

How to evaluate the limit $\frac{\mathrm{sin}\left(5x\right)}{x}$ as x approaches 0?

How to find derivatives of parametric functions?

What is the antiderivative of $-5{e}^{x-1}$?

How to evaluate: indefinite integral $\frac{1+x}{1+{x}^{2}}dx$?