integrateByParts

Create a symbolic expression F that is the integral of a product of functions.

syms u(x) v(x)
F = int(u*diff(v))

F(x) = 
$$\int u\left(x\right)\hspace{0.17em}\frac{\partial }{\partial x}\mathrm{ }v\left(x\right)\mathrm{ }\mathrm{d}x$$

Apply integration by parts to F.

g = integrateByParts(F,diff(u))

g = 
$$u\left(x\right)\hspace{0.17em}v\left(x\right)-\int v\left(x\right)\hspace{0.17em}\frac{\partial }{\partial x}\mathrm{ }u\left(x\right)\mathrm{ }\mathrm{d}x$$

Apply integration by parts to the integral $\int {\mathit{x}}^2{\mathit{e}}^{\mathit{x}}\mathit{dx}$.

Define the integral using the int function. Show the result without evaluating the integral by setting the 'Hold' option to true.

syms x
F = int(x^2*exp(x),'Hold',true)

F = 
$$\int x^2\hspace{0.17em}{\mathrm{e}}^x\mathrm{ }\mathrm{d}x$$

To show the steps of integration, apply integration by parts to F and use exp(x) as the differential to be integrated.

G = integrateByParts(F,exp(x))

G = 
$$x^2\hspace{0.17em}{\mathrm{e}}^x-\int 2\hspace{0.17em}x\hspace{0.17em}{\mathrm{e}}^x\mathrm{ }\mathrm{d}x$$

H = integrateByParts(G,exp(x))

H = 
$$x^2\hspace{0.17em}{\mathrm{e}}^x-2\hspace{0.17em}x\hspace{0.17em}{\mathrm{e}}^x+\int 2\hspace{0.17em}{\mathrm{e}}^x\mathrm{ }\mathrm{d}x$$

Evaluate the integral in H by using the release function to ignore the 'Hold' option.

F1 = release(H)

F1 = $$2\hspace{0.17em}{\mathrm{e}}^x+x^2\hspace{0.17em}{\mathrm{e}}^x-2\hspace{0.17em}x\hspace{0.17em}{\mathrm{e}}^x$$

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

F2 = int(x^2*exp(x))

F2 = $${\mathrm{e}}^x\hspace{0.17em}\left(x^2-2\hspace{0.17em}x+2\right)$$

Apply integration by parts to the integral $\int {\mathit{e}}^{\mathit{ax}}\sin \left(\mathit{bx}\right)\mathit{dx}$.

Define the integral using the int function. Show the integral without evaluating it by setting the 'Hold' option to true.

syms x a b
F = int(exp(a*x)*sin(b*x),'Hold',true)

F = 
$$\int {\mathrm{e}}^{a\hspace{0.17em}x}\hspace{0.17em}\sin \left(b\hspace{0.17em}x\right)\mathrm{ }\mathrm{d}x$$

To show the steps of integration, apply integration by parts to F and use ${\mathit{u}}^{\prime }\left(\mathit{x}\right)={\mathit{e}}^{\mathit{ax}}$ as the differential to be integrated.

G = integrateByParts(F,exp(a*x))

G = 
$$\frac{{\mathrm{e}}^{a\hspace{0.17em}x}\hspace{0.17em}\sin \left(b\hspace{0.17em}x\right)}{a}-\int \frac{b\hspace{0.17em}{\mathrm{e}}^{a\hspace{0.17em}x}\hspace{0.17em}\cos \left(b\hspace{0.17em}x\right)}{a}\mathrm{ }\mathrm{d}x$$

Evaluate the integral in G by using the release function to ignore the 'Hold' option.

F1 = release(G)

F1 = 
$$\frac{{\mathrm{e}}^{a\hspace{0.17em}x}\hspace{0.17em}\sin \left(b\hspace{0.17em}x\right)}{a}-\frac{b\hspace{0.17em}{\mathrm{e}}^{a\hspace{0.17em}x}\hspace{0.17em}\left(a\hspace{0.17em}\cos \left(b\hspace{0.17em}x\right)+b\hspace{0.17em}\sin \left(b\hspace{0.17em}x\right)\right)}{a\hspace{0.17em}\left(a^2+b^2\right)}$$

Simplify the result.

F2 = simplify(F1)

F2 = 
$$-\frac{{\mathrm{e}}^{a\hspace{0.17em}x}\hspace{0.17em}\left(b\hspace{0.17em}\cos \left(b\hspace{0.17em}x\right)-a\hspace{0.17em}\sin \left(b\hspace{0.17em}x\right)\right)}{a^2+b^2}$$