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The Rule: . arctan. OR . arctan. Putting everything together: . Use this rule when you have a fraction of the form: where the polynomial in the denominator does . not factor and the fraction is not in the correct form to turn into an . Example: evaluate . From the example above we know that .
Rule: If p-1∈ℤ+ ∧ c2 d2 +e2 ⩵ 0∧ (m q)∈ℤ∧ q≠ -1, let u→∫(f x)m (d+e x)q ⅆx, then f x m (d + e x) q (a + b ArcTan[c x]) p ⅆ x
7.3 CALCULUS WITH THE INVERSE TRIGONOMETRIC FUNCTIONS. The three previous sections introduced the ideas of one–to–one functions and inverse functions and used. those ideas to define arcsine, arctangent, and the other inverse trigonometric functions. Section 7.3 presents. the calculus of inverse trigonometric functions.
Rule: If p∈ℤ+, then a + b ArcTan c xn p x ⅆx 1 n Subst (a + b ArcTan[c x])p x ⅆx, x, xn Program code: Int[(a_.+b_.*ArcTan[c_.*x_^n_])^p_./x_,x_Symbol] := 1/n*Subst[Int[(a+b*ArcTan[c*x])^p/x,x],x,x^n] /; FreeQ[{a,b,c,n},x] && IGtQ[p,0] Int[(a_.+b_.*ArcCot[c_.*x_^n_])^p_./x_,x_Symbol] :=
The Mathematical Functions Site
www.mathportal.org 5. Integrals of Trig. Functions ∫sin cosxdx x= − ∫cos sinxdx x= − sin sin22 1 2 4 x ∫ xdx x= − cos sin22 1 2 4 x ∫ xdx x= + sin cos cos3 31 3 ∫ xdx x x= − cos sin sin3 31 3 ∫ xdx x x= − ln tan sin 2 dx x xdx x ∫=
Inverse Trigonometric Functions: Integration. Integrate functions whose antiderivatives involve inverse trigonometric functions. Use the method of completing the square to integrate a function. Review the basic integration rules involving elementary functions.
