Define a univariate expression.

syms x
expr = -2*x/(1+x^2)^2;

Find the indefinite integral of the univariate expression.

F = int(expr)

F = 
$$\frac{1}{x^2+1}$$

Define a multivariate function with variables x and z.

syms x z
f(x,z) = x/(1+z^2);

Find the indefinite integrals of the multivariate expression with respect to the variables x and z.

Fx = int(f,x)

Fx(x, z) = 
$$\frac{x^2}{2\hspace{0.17em}\left(z^2+1\right)}$$

Fz = int(f,z)

Fz(x, z) = $$x\hspace{0.17em}\mathrm{atan}\left(z\right)$$

If you do not specify the integration variable, then int uses the first variable returned by symvar as the integration variable.

var = symvar(f,1)

var = $$x$$

F = int(f)

F(x, z) = 
$$\frac{x^2}{2\hspace{0.17em}\left(z^2+1\right)}$$

Integrate a symbolic expression from 0 to 1.

syms x
expr = x*log(1+x);
F = int(expr,[0 1])

F = 
$$\frac{1}{4}$$

Integrate another expression from sin(t) to 1.

syms t
F = int(2*x,[sin(t) 1])

F = $${\cos \left(t\right)}^2$$

When int cannot compute the value of a definite integral, numerically approximate the integral by using vpa.

syms x
f = cos(x)/sqrt(1 + x^2);
Fint = int(f,x,[0 10])

Fint = 
$${\int }_0^{10}\frac{\cos \left(x\right)}{\sqrt{x^2+1}}\mathrm{ }\mathrm{d}x$$

Fvpa = vpa(Fint)

Fvpa = $$0.37570628299079723478493405557162$$

To approximate integrals directly, use vpaintegral instead of vpa. The vpaintegral function is faster and provides control over integration tolerances.

Fvpaint = vpaintegral(f,x,[0 10])

Fvpaint = $$0.375706$$

Define a symbolic matrix containing four expressions as its elements.

syms a x t z
M = [exp(t) exp(a*t); sin(t) cos(t)]

M = 
$$\left(\begin{array}{cc}{\mathrm{e}}^t& {\mathrm{e}}^{a\hspace{0.17em}t}\\ \sin \left(t\right)& \cos \left(t\right)\end{array}\right)$$

Find indefinite integrals of the matrix element-wise.

F = int(M,t)

F = 
$$\left(\begin{array}{cc}{\mathrm{e}}^t& \frac{{\mathrm{e}}^{a\hspace{0.17em}t}}{a}\\ -\cos \left(t\right)& \sin \left(t\right)\end{array}\right)$$

Define a symbolic function and compute its indefinite integral.

syms f(x)
f(x) = acos(cos(x));
F = int(f,x)

F(x) = 
$$x\hspace{0.17em}\mathrm{acos}\left(\cos \left(x\right)\right)-\frac{x^2}{2\hspace{0.17em}\mathrm{sign}\left(\sin \left(x\right)\right)}$$

By default, int uses strict mathematical rules. These rules do not let int rewrite acos(cos(x)) as x.

If you want a simple practical solution, set 'IgnoreAnalyticConstraints' to true.

F = int(f,x,'IgnoreAnalyticConstraints',true)

F(x) = 
$$\frac{x^2}{2}$$

Define a symbolic expression ${\mathit{x}}^{\mathit{t}}$ and compute its indefinite integral with respect to the variable $$x$$.

syms x t
F = int(x^t,x)

F = 
$$\{\begin{array}{cl}\log \left(x\right)& \text{ if }t=-1\\ \frac{x^{t+1}}{t+1}& \text{ if }t\ne -1\end{array}$$

By default, int returns the general results for all values of the other symbolic parameter t. In this example, int returns two integral results for the case $$t=-1$$ and $$t\ne -1$$.

To ignore special cases of parameter values, set 'IgnoreSpecialCases' to true. With this option, int ignores the special case $$t=-1$$ and returns the solution for $$t\ne -1$$.

F = int(x^t,x,'IgnoreSpecialCases',true)

F = 
$$\frac{x^{t+1}}{t+1}$$

Define a symbolic function $$f(x)=1/(x-1)$$ that has a pole at $$x=1$$.

syms x
f(x) = 1/(x-1)

f(x) = 
$$\frac{1}{x-1}$$

Compute the definite integral of this function from $$x=0$$ to $$x=2$$. Since the integration interval includes the pole, the result is not defined.

F = int(f,[0 2])

F = $$\mathrm{NaN}$$

However, the Cauchy principal value of the integral exists. To compute the Cauchy principal value of the integral, set 'PrincipalValue' to true.

F = int(f,[0 2],'PrincipalValue',true)

F = $$0$$

Find the integral of $\int \mathit{x}{\mathit{e}}^{\mathit{x}}\mathit{dx}$.

Define the integral without evaluating it by setting the 'Hold' option to true.

syms x g(y)
F = int(x*exp(x),'Hold',true)

F = 
$$\int x\hspace{0.17em}{\mathrm{e}}^x\mathrm{ }\mathrm{d}x$$

You can apply integration by parts to F by using the integrateByParts function. Use exp(x) as the differential to be integrated.

G = integrateByParts(F,exp(x))

G = 
$$x\hspace{0.17em}{\mathrm{e}}^x-\int {\mathrm{e}}^x\mathrm{ }\mathrm{d}x$$

To evaluate the integral in G, use the release function to ignore the 'Hold' option.

Gcalc = release(G)

Gcalc = $$x\hspace{0.17em}{\mathrm{e}}^x-{\mathrm{e}}^x$$

Compare the result to the integration result returned by int without setting the 'Hold' option.

Fcalc = int(x*exp(x))

Fcalc = $${\mathrm{e}}^x\hspace{0.17em}\left(x-1\right)$$

If int cannot compute a closed form of an integral, then it returns an unresolved integral.

syms f(x)
f(x) = sin(sinh(x));
F = int(f,x)

F(x) = 
$$\int \sin \left(\sinh \left(x\right)\right)\mathrm{ }\mathrm{d}x$$

You can approximate the integrand function $$f(x)$$ as polynomials by using the Taylor expansion. Apply taylor to expand the integrand function $$f(x)$$ as polynomials around $$x=0$$. Compute the integral of the approximated polynomials.

fTaylor = taylor(f,x,'ExpansionPoint',0,'Order',10)

fTaylor(x) = 
$$\frac{x^9}{5670}-\frac{x^7}{90}-\frac{x^5}{15}+x$$

Fapprox = int(fTaylor,x)

Fapprox(x) = 
$$\frac{x^{10}}{56700}-\frac{x^8}{720}-\frac{x^6}{90}+\frac{x^2}{2}$$