Calculator will approximate integral using trapezoidal rule.

In general, you can skip multiplication sign, so `5x` is equivalent to `5*x`

In general, you can skip parentheses, but be very careful: e^3x is `e^3x` and e^(3x) is `e^(3x)`

Also be careful when you write fractions: 1/x^2 ln(x) is `1/x^2 ln(x)` and 1/(x^2 ln(x)) is `1/(x^2 ln(x))`

If you skip parentheses or multiplication sign, type at least whitespace, i.e. write sin x (or even better sin(x)) instead of sinx

Sometimes I see expressions like tan^2xsec^3x: this will be parsed as `tan^(2*3)(x sec(x))`. To get `tan^2(x)sec^3(x)`, use parentheses: tan^2(x)sec^3(x)

Similarly tanxsec^3x will be parsed as `tan(xsec^3(x))`. To get `tan(x)sec^3(x)`, use parentheses: tan(x)sec^3(x)

From table below you can notice, that sech is not supported, but you can still enter it using identity `sech(x)=1/cosh(x)`

If you get an error, double check your expression, add parentheses and multiplication signs, where needed, and consult the table below

All suggestions and improvements are welcomed. Leave them in comments

Following table contains supported operations and functions:

Type

Get

Constants

e

e

pi

`pi`

i

i (imaginary unit)

Operations

a+b

a+b

a-b

a-b

a*b

`a*b`

a^b, a**b

`a^b`

sqrt(x), x^(1/2)

`sqrt(x)`

cbrt(x), x^(1/3)

`root(3)(x)`

root(x,n), x^(1/n)

`root(n)(x)`

x^(a/b)

`x^(a/b)`

Functions

e^x

`e^x`

ln(x), log(x)

ln(x)

ln(x)/ln(a)

`log_a(x)`

Trigonometric Functions

sin(x)

sin(x)

cos(x)

cos(x)

tan(x)

tan(x), tg(x)

cot(x)

cot(x), ctg(x)

sec(x)

sec(x)

csc(x)

csc(x), cosec(x)

Inverse Trigonometric Functions

asin(x), arcsin(x), sin^-1(x)

asin(x)

acos(x), arccos(x), cos^-1(x)

acos(x)

atan(x), arctan(x), tan^-1(x)

atan(x)

acot(x), arccot(x), cot^-1(x)

acot(x)

asec(x), arcsec(x), sec^-1(x)

asec(x)

acsc(x), arccsc(x), csc^-1(x)

acsc(x)

Hyperbolic Functions

sinh(x)

sinh(x)

cosh(x)

cosh(x)

tanh(x)

tanh(x)

coth(x)

coth(x)

1/cosh(x)

sech(x)

1/sinh(x)

csch(x)

Inverse Hyperbolic Functions

asinh(x), arcsinh(x), sinh^-1(x)

asinh(x)

acosh(x), arccosh(x), cosh^-1(x)

acosh(x)

atanh(x), arctanh(x), tanh^-1(x)

atanh(x)

acoth(x), arccoth(x), cot^-1(x)

acoth(x)

acosh(1/x)

asech(x)

asinh(1/x)

acsch(x)

Solution

Your input: approximate integral $$$\displaystyle{\int_{0}^{1}\sqrt{\sin^{3}{\left (x \right )} + 1}\ dx}$$$ using $$$n=5$$$ rectangles.

Trapezoidal rule states that $$$\displaystyle{\int_{a}^{b}f(x)dx}\approx\frac{\Delta{x}}{2}\left(f(x_0)+2f(x_1)+2f(x_2)+...+2f(x_{n-1})+f(x_n)\right)$$$, where $$$\displaystyle{\Delta{x}=\frac{b-a}{n}}$$$.

Divide interval $$$\left[0,1\right]$$$ into $$$n=5$$$ subintervals of length $$$\Delta{x}=\frac{1}{5}$$$ with the following endpoints: $$$a=0, \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5}, 1=b$$$.

Now, we just evaluate function at those endpoints:

Finally, just sum up above values and multiply by $$$\displaystyle\frac{\Delta{x}}{2}=\frac{1}{10}$$$: $$$\frac{1}{10}(1+2.00782606791279+2.05820697233265+2.17257446116512+2.34021475342487+1.26325897447473)=1.08420812293102$$$