# Tag Info

## Hot answers tagged calculus

2 votes
Accepted

### Need help with the Inverse Laplace transform

Convolution theorem: $$\mathcal L(f(t)*f(t))=(\bar f(s))^2$$ Here $$\bar f(s)=4/(s^2+4)$$ So \begin{aligned} f(t)=\mathcal{L}^{-1}(\bar{f}(s)) & =\mathcal{L}^{-1}\left(\frac{4}{s^2+4}\right) \\ &...
• 194
2 votes

### definite integral $\int_{0}^{\frac{\pi}{4}} \frac{\sin^2x\cos^2x}{\sin^3x+\cos^3x}dx$

Note that \begin{align} I=&\int_{0}^{\frac{\pi}{4}} \frac{\sin^2x\cos^2x}{\sin^3x+\cos^3x}dx\\ =&\int_{0}^{\frac{\pi}{4}} \frac{(\sin x\cos x)^2(\sin x+\cos x)}{(\sin x+\cos x)^2(\sin^2x-\sin ...
• 97.5k
1 vote

### Help request for calculating an integral

Alternatively, let $x=\frac t{\sqrt{1+t^2}}$ \begin{align} \int \frac{2\sqrt{1- x^2}}{2 x\sqrt{1- x^2}+ 5}dx =& \int \frac{2}{(1+t^2)(5t^2+2t +5)}dt\\ =&\ \int \frac{t+\frac25}{t^2+\frac25 t +...
• 97.5k
1 vote

• 19.5k
1 vote

1 vote

### What is $\int (y'(x))^2 dx$?

It is true that you cannot do this integral without knowing what $y$ is. But we can simplify it: $$\int (y'(x))^2 dx = \int y'(x)y'(x) dx=y(x)y'(x)-\int y''(x)y(x)dx$$ There isn't much we can do ...
• 567
1 vote

### Find Maximum and Minimum distance from origin to $f(x,y)$ using the Lagrange method.

You've done fine to conclude that $x^2=y^2.$ That means $x=y$ or that $x=-y.$ Plug each of those into your curve's equation. You should find four distinct points of two distinct distances from the ...
• 103k
1 vote
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• 35.9k
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### Find $\int_{0}^{\pi/2} \frac{\mathrm{d}\theta}{(\sin\theta+1)^2}$.

Use $u=x+\sqrt{1+x^2}$(for the original integral, this substitution is also not a bad choice), then $$x=\frac{1}{2}(u-1/u),\quad {\rm d}x=\frac12(1+1/u^2){\rm d}u.$$ So \int_{0}^\infty \frac{{\rm d}...
• 7,245

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