Integration in Maple Maple makes it easy to integrate functions of all types. We will demonstrate how to do both indefinite and definite integrals.

The key command for integration is int. Let us start by looking at indefinite integrals. We demonstrate with the following two examples. Suppose we want to find an anti-derivative of 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

Suppose we want to find an anti-derivative of 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. We can do this easily, just enter f := x -> x*exp(-x);

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int(f(x),x);

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The syntax of the int command is int(expression, variable). When written with an upper case I, the command Int is the inert version of int, meaning that it displays the integral, but does not compute anything: Int(f(x),x);

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It is recommended to use Int along with value. Int(f(x),x); value(%);

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This example demonstrates how to integrate the function 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 g := x -> (x^6 + x^4 + x^2 + 1) / ( (x-1)^2 * (x^2 + 3)^2 );

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int(g(x), x);

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Int(g(x), x); value(%);

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Maple can't find every integral; if it gets too hard, maple will just repeat the question. h := x -> exp(-x)*sin(sqrt(1+x^2));

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Int(h(x),x); value(%);

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This is maple's way to say "I don't know".

Maple omits the constant of integration C. This may occasionally lead to seemingly contradictory results. Let's say we want to integrate 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 . Direct computation yields y := (x-1)^2; Int(y,x); Y := value(%);

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But when we expand y first, we obtain a different answer. z := expand(y); Int(z,x); Z := value(%);

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Z-Y;

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simplify(%);

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We see that the two answers differ by the constant 1/3. This confirms that anti-derivatives differ by a constant, and it also teaches us to be attentive in the use of Maple.

Maple can quickly evaluate definite integrals. We demonstrate with the following examples. Example: Integrate 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 from 0 to 4.

Suppose we want to compute 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 We can do this by entering the following commands f := x -> exp(x)*sin(2*x);

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int(f(x), x = 0..Pi); evalf(%);

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We just get the result. Again, it is preferable to use the Int and value combination: Int(f(x), x = 0..Pi); value(%); evalf(%);

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Suppose we want to compute the definite integral 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 This can be accomplished with the following commands g := x -> 1/(x^4*sqrt(49-x^2));

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int(g(x), x = 1..2); evalf(%);

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Int(g(x), x = 1..2); value(%); evalf(%);

NiMtJSRJbnRHNiQqJiIiIiIiIiomKSUieEciIiUiIiIpLCYiI1wiIiIqJCklInhHIiIjIiIiISIiIyIiIiIiIyIiIiEiIi8lInhHOyIiIiIiIw== NiMsJiooIiNvIiIiIiUsQyEiIikiIiQjIiIiIiIjIiIiIiIiKigiI2QiIiIiJjMjPiEiIikiIiYjIiIiIiIjIiIiISIi NiMkIitmcig9QyUhIzY=

Maple can't solve every integral; if it gets too hard, maple will just repeat the question, as we saw above. h(x) := exp(-x)*sin(sqrt(1+x^2));

NiM+LSUiaEc2IyUieEcqJi0lJGV4cEc2IywkJSJ4RyEiIiIiIi0lJHNpbkc2IyokKSwmIiIiIiIiKiQpJSJ4RyIiIyIiIiIiIiMiIiIiIiMiIiIiIiI=

Int(h(x), x = 0..3); value(%);

NiMtJSRJbnRHNiQqJi0lJGV4cEc2IywkJSJ4RyEiIiIiIi0lJHNpbkc2IyokKSwmIiIiIiIiKiQpJSJ4RyIiIyIiIiIiIiMiIiIiIiMiIiIiIiIvJSJ4RzsiIiEiIiQ= NiMtJSRpbnRHNiQqJi0lJGV4cEc2IywkJSJ4RyEiIiIiIi0lJHNpbkc2IyokKSwmIiIiIiIiKiQpJSJ4RyIiIyIiIiIiIiMiIiIiIiMiIiIiIiIvJSJ4RzsiIiEiIiQ=

But we can always ask for a numerical approximation. evalf(Int(h(x), x = 0..3));

NiMkIitmOiV5RikhIzU=

1, Compute the following antiderivatives a. 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 b. 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 c. 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 2. Compute the following definite integrals. a. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2KS1JKG1zdWJzdXBHRiQ2Jy1JI21vR0YkNi9RJiZpbnQ7RicvJStmb3JlZ3JvdW5kR1EuWzE0NCwxNDQsMTQ0XUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy9JK21zZW1hbnRpY3NHRiRRJmluZXJ0RicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkMvJSlzdHJldGNoeUdGQy8lKnN5bW1ldHJpY0dGQy8lKGxhcmdlb3BHRkMvJS5tb3ZhYmxlbGltaXRzR0ZDLyUnYWNjZW50R0ZDLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUi1JI21uR0YkNiRRIjBGJ0Y7LUZWNiRRIjFGJ0Y7LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjJGJy8lL3N1YnNjcmlwdHNoaWZ0R0ZYRistRiM2Ji1JJW1zdXBHRiQ2JS1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvRjxRJ2l0YWxpY0YnLUZWNiRRIjRGJ0Y7L0ZnbkZYLUY1Ni1RMSZJbnZpc2libGVUaW1lcztGJ0Y7RkFGREZGRkhGSkZMRk5GUEZTLUYjNiUtRiw2JVEjbG5GJy9GZG9GQ0Y7LUY1Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRjtGQUZERkZGSEZKRkxGTkZQRlMtSShtZmVuY2VkR0YkNiQtRiM2JUYrLUYjNiVGYG8tRjU2LVEiK0YnRjtGQUZERkZGSEZKRkxGTi9GUVEsMC4yMjIyMjIyZW1GJy9GVEZjcUZZRitGO0YrRistSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuM2VtRicvJSZkZXB0aEdGanEvJSpsaW5lYnJlYWtHUSVhdXRvRictRjU2L1EwJkRpZmZlcmVudGlhbEQ7RidGOEY7Rj5GQUZERkZGSEZKRkxGTkZQRlNGYG9GKw== b. 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 c. 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