Applications of Integration

The speed function over time

The acceleration function over time

The velocity function squared over time

The original position functions over time

Improper integral.

Integral involving partial fractions.

Definite integral.

Indefinite integral.

$\int_{}^{} \sqrt{(\sinh(t))^2 + (\cosh(t))^2} dt$

\int_{}^{} \sqrt{(\sinh(\log)) + \cos(n)p(t)) dt

$\int_{}^{} \cosh(t) \tan(dt) dT$

$\int_{}^{} \sqrt{(\cosh(\tan(t))^2 + (\sinh(t))^2} dt$

There must exist two points in [c,d] where g(t) has equal values.

$g(t)$ must have an inverse on [c,d].

$g(t)$ must also be differentiable on [c,d].

$g(t)$'s range must be all real numbers.

$\int (1 + (e^{2x})^2) , dx$

$\int (1 + (e^{2x})^2)^{1/2} , dx$

$\int (1 + (e^x)^2)^{1/2} , dx$

$\int (1 + (e^x)^2) , dx$

$\int (1 + (\ln(x))^2) , dx$

$\int (1 + (\ln(x))^2)^{1/2} , dx$

$\int (1 + (\ln(e))^2) , dx$

$\int (1 + (\ln(e))^2)^{1/2} , dx$

The curve must be defined on a closed interval

The curve must be continuous

The curve must be bounded

The curve must be differentiable

$r = x + y$

$y = r + x$

$r = |x|$

$x = y - r$

Sum up all infinitesimal distances $(dx)^{-e^{(-x)}}$ from $x=0$ to infinity

Calculate $\frac{1}{e^{-x}}$ evaluated from $x=0$ to $+\infty$

Evaluate $\lim_{a \to +\infty} \int_0^{a} e^{-x} dx$

Evaluate $\lim_{a \to +\infty} \int_0^{a} \sqrt{1+e^{-2x}} dx$

Find h(4) - h(0).

Calculate $\frac{dh}{dx}$ at x=4 and x=0 and subtract them.

Integrate $h(x)$ from x=0 to x=4 directly without any modifications.

Evaluate $\int_0^4 \sqrt{1+(h'(x))^2},dx$ where $h'(x)$ is its derivative with respect to x.