release

Define a symbolic call to an integral $\int \cos \left(\mathit{x}\right)\mathit{dx}$ without evaluating it. Set the 'Hold' option to true when defining the integral using the int function.

syms x
F = int(cos(x),'Hold',true)

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

Use release to evaluate the integral by ignoring the 'Hold' option.

G = release(F)

G = $$\sin \left(x\right)$$

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

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