changeIntegrationVariable

Apply a change of variable to the definite integral ${\int }_{\mathit{a}}^{\mathit{b}}\mathit{f}\left(\mathit{x}+\mathit{c}\right)\mathit{dx}$.

Define the integral.

syms f(x) y a b c
F = int(f(x+c),a,b)

F = 
$${\int }_a^{b}f\left(c+x\right)\mathrm{ }\mathrm{d}x$$

Change the variable $\mathit{x}+\mathit{c}$ in the integral to $\mathit{y}$.

G = changeIntegrationVariable(F,x+c,y)

G = 
$${\int }_{a+c}^{b+c}f\left(y\right)\mathrm{ }\mathrm{d}y$$

Find the integral of $\int \cos \left(\log \left(\mathit{x}\right)\right)\mathit{dx}$ using integration by substitution.

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

syms x t
F = int(cos(log(x)),'Hold',true)

F = 
$$\int \cos \left(\log \left(x\right)\right)\mathrm{ }\mathrm{d}x$$

Substitute the expression log(x) with t.

G = changeIntegrationVariable(F,log(x),t) 

G = 
$$\int {\mathrm{e}}^t\hspace{0.17em}\cos \left(t\right)\mathrm{ }\mathrm{d}t$$

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

H = release(G)

H = 
$$\frac{{\mathrm{e}}^t\hspace{0.17em}\left(\cos \left(t\right)+\sin \left(t\right)\right)}{2}$$

Restore log(x) in place of t.

H = simplify(subs(H,t,log(x)))

H = 
$$\frac{\sqrt{2}\hspace{0.17em}x\hspace{0.17em}\sin \left(\frac{\pi }{4}+\log \left(x\right)\right)}{2}$$

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

Fcalc = int(cos(log(x)))

Fcalc = 
$$\frac{\sqrt{2}\hspace{0.17em}x\hspace{0.17em}\sin \left(\frac{\pi }{4}+\log \left(x\right)\right)}{2}$$

Find the closed-form solution of the integral $\int \mathit{x}\tan \left(\log \left(\mathit{x}\right)\right)\mathit{dx}$.

Define the integral using the int function.

syms x
F = int(x*tan(log(x)),x)

F = 
$$\int x\hspace{0.17em}\tan \left(\log \left(x\right)\right)\mathrm{ }\mathrm{d}x$$

The int function cannot find the closed-form solution of the integral.

Substitute the expression log(x) with t. Apply integration by substitution.

syms t
G = changeIntegrationVariable(F,log(x),t)

G = 
$$\frac{{\mathrm{e}}^{2\hspace{0.17em}t}\hspace{0.17em}{}_2{\mathrm{F}\text{<a href="matlab:doc symbolic/hypergeom">hypergeom</a>}}_1\left(1,-\mathrm{i};\mathrm{ }1-\mathrm{i};\mathrm{ }-{\mathrm{e}}^{2\hspace{0.17em}t\hspace{0.17em}\mathrm{i}}\right)\hspace{0.17em}\mathrm{i}}{2}+{\mathrm{e}}^{t\hspace{0.17em}\left(2+2\hspace{0.17em}\mathrm{i}\right)}\hspace{0.17em}{}_2{\mathrm{F}\text{<a href="matlab:doc symbolic/hypergeom">hypergeom</a>}}_1\left(1,1-\mathrm{i};\mathrm{ }2-\mathrm{i};\mathrm{ }-{\mathrm{e}}^{2\hspace{0.17em}t\hspace{0.17em}\mathrm{i}}\right)\hspace{0.17em}\left(-\frac{1}{4}-\frac{1}{4}\hspace{0.17em}\mathrm{i}\right)$$

The closed-form solution is expressed in terms of hypergeometric functions. For more details on hypergeometric functions, see hypergeom.

Compute the integral ${\int }_0^1{\mathit{e}}^{\sqrt{\sin \left(\mathit{x}\right)}}\mathit{dx}$ numerically with high precision.

Define the integral ${\int }_0^1{\mathit{e}}^{\sqrt{\sin \left(\mathit{x}\right)}}\mathit{dx}$. A closed-form solution to the integral does not exist.

syms x
F = int(exp(sqrt(sin(x))),x,0,1)

F = 
$${\int }_0^1{\mathrm{e}}^{\sqrt{\sin \left(x\right)}}\mathrm{ }\mathrm{d}x$$

You can use vpa to compute the integral numerically to 10 significant digits.

F10 = vpa(F,10)

F10 = $$1.944268879$$

Alternatively, you can use the vpaintegral function and specify the relative error tolerance.

Fvpa = vpaintegral(exp(sqrt(sin(x))),x,0,1,'RelTol',1e-10)

Fvpa = $$1.944268879$$

The vpa function cannot find the numerical integration to 70 significant digits, and it returns the unevaluated integral in the form of vpaintegral.

F70 = vpa(F,70)

F70 = $$1.944268879138581167466225761060083173280747314051712224507065962575967$$

To find the numerical integration with high precision, you can perform a change of variable. Substitute the expression $\sqrt{\sin \left(\mathit{x}\right)}$ with $\mathit{t}$. Compute the integral numerically to 70 significant digits.

syms t;
G = changeIntegrationVariable(F,sqrt(sin(x)),t)

G = 
$${\int }_0^{\sqrt{\sin \left(1\right)}}\frac{2\hspace{0.17em}t\hspace{0.17em}{\mathrm{e}}^t}{\sqrt{1-t^4}}\mathrm{ }\mathrm{d}t$$

G70 = vpa(G,70)

G70 = $$1.944268879138581167466225761060083173280747314051712224507065962575967$$