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Evaluate ∫_0^41/(x^2+2x+3)dx (a) (√2)/2arctan(2√2) (b) (√2)/2[arctan(5/(√2))-a…

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Wize University Calculus 1 Textbook > Integration Techniques

Trigonometric Substitution

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Evaluate ∫041x2+2x+3dx\int_0^4\frac{1}{x^2+2x+3}dx

(a) 22arctan⁡(22)\frac{\sqrt2}{2}\arctan({2\sqrt2}) (b) 22[arctan⁡(52)−arctan⁡(12)]\frac{\sqrt{2}}{2}\left[\arctan\left(\frac{5}{\sqrt{2}}\right)-\arctan\left(\frac{1}{\sqrt{2}}\right)\right] (c) 2[arcsin⁡(52)+arctan⁡(12)]2\left[\arcsin\left(\frac{5}{\sqrt{2}}\right)+\arctan\left(\frac{1}{\sqrt{2}}\right)\right]
More Trigonometric Substitution Questions:
Trigonometric Substitution
Find ∫1x2−2x−3  ⁣dx{\displaystyle\int}\frac{1}{\sqrt{x^2-2x-3}}\de{x}.
Practice Question: Trig Sub
Find ∫1x2−2x−3  ⁣dx{\displaystyle\int}\frac{1}{\sqrt{x^2-2x-3}}\de{x}.
Practice Question: Trig Sub
Find ∫1x2−2x−3  ⁣dx{\displaystyle\int}\frac{1}{\sqrt{x^2-2x-3}}\de{x}.
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Evaluate ∫1x2x2−1 dx\displaystyle\int\frac{1}{x^2\sqrt{x^2-1}} \ dx
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Evaluate ∫dxx2+4x+8\displaystyle\int\frac{dx}{\sqrt{x^2+4x+8}}
Trigonometric Substitution: Complete the Square, Sec
Evaluate ∫1x2x2−1 dx\displaystyle\int\frac{1}{x^2\sqrt{x^2-1}} \ dx
Trigonometric Substitution: Complete the Square, Sec
Evaluate ∫1x2x2−1 dx\displaystyle\int\frac{1}{x^2\sqrt{x^2-1}} \ dx
Practice Question: Trig Sub
Practice Question
Find ∫1x2−2x−3  ⁣dx{\displaystyle\int}\frac{1}{\sqrt{x^2-2x-3}}\de{x}.
Trigonometric Substitution: Complete the Square, Sec
Evaluate ∫1x2x2−1 dx\displaystyle\int\frac{1}{x^2\sqrt{x^2-1}} \ dx
Trigonometric Substitution: Complete the Square, Sec
Evaluate ∫1x2x2−1 dx\displaystyle\int\frac{1}{x^2\sqrt{x^2-1}} \ dx
Trigonometric Substitution: Complete the Square, Sec
Evaluate ∫1x2x2−1 dx\displaystyle\int\frac{1}{x^2\sqrt{x^2-1}} \ dx
Trigonometric Substitution: Complete the Square, Sec
Evaluate ∫1x2x2−1 dx\displaystyle\int\frac{1}{x^2\sqrt{x^2-1}} \ dx
Trigonometric Substitution: Complete the Square, Sec
Evaluate ∫1x2x2−1 dx\displaystyle\int\frac{1}{x^2\sqrt{x^2-1}} \ dx
Trigonometric Substitution: Complete the Square, Sec
Evaluate ∫1x2x2−1 dx\displaystyle\int\frac{1}{x^2\sqrt{x^2-1}} \ dx
Trigonometric Substitution: Complete the Square, Sec
Evaluate ∫1x2x2−1 dx\displaystyle\int\frac{1}{x^2\sqrt{x^2-1}} \ dx
Trigonometric Substitution: Complete the Square, Sec
Evaluate ∫1x2x2−1 dx\displaystyle\int\frac{1}{x^2\sqrt{x^2-1}} \ dx
Trigonometric Substitution: Complete the Square, Sec
Evaluate ∫1x2x2−1 dx\displaystyle\int\frac{1}{x^2\sqrt{x^2-1}} \ dx
Trigonometric Substitution: Complete the Square, Sec
Evaluate ∫1x2x2−1 dx\displaystyle\int\frac{1}{x^2\sqrt{x^2-1}} \ dx
Trigonometric Substitution: Complete the Square, Sec
Evaluate ∫1x2x2−1 dx\displaystyle\int\frac{1}{x^2\sqrt{x^2-1}} \ dx
Trigonometric Substitution: Complete the Square, Sec
Evaluate ∫1x2x2−1 dx\displaystyle\int\frac{1}{x^2\sqrt{x^2-1}} \ dx
Trigonometric Substitution: Complete the Square, Sec
Evaluate ∫1x2x2−1 dx\displaystyle\int\frac{1}{x^2\sqrt{x^2-1}} \ dx
Practice: Complete the Square, Sec
Q:\bf{Q:}Evaluate ∫x2−4x+2 dx\displaystyle\int\sqrt{x^2-4x+2} \ dx
Trigonometric Substitution: Complete the Square, Tan
Evaluate ∫dxx2+4x+8\displaystyle\int\frac{dx}{\sqrt{x^2+4x+8}}
Practice: Use Sin
Q:\bf{Q:} Evaluate ∫x+29−x2dx\displaystyle\int\frac{x+2}{\sqrt{9-x^2}}dx
Integration Medley
Practice Question
Evaluate ∫1(x2−6x+11)2  ⁣dx{\displaystyle\int} \frac{1}{(x^2-6x+11)^2}\de{x}.
Evaluate ∫99x2−2x+6dx\int_{ }^{ }\frac{9}{9x^2-2x+6}dx
Evaluate ∫x2(x2+9)32dx\int_{ }^{ }\frac{x^2}{\left(x^2+9\right)^{\frac{3}{2}}}dx
Evaluate ∫041x2+2x+3dx\int_0^4\frac{1}{x^2+2x+3}dx
Integration Medley
Practice Question
Evaluate ∫1(x2−6x+11)2  ⁣dx{\displaystyle\int} \frac{1}{(x^2-6x+11)^2}\de{x}.
Practice Question: Trig Sub
Practice Question
Find ∫1x2−2x−3  ⁣dx{\displaystyle\int}\frac{1}{\sqrt{x^2-2x-3}}\de{x}.
Trigonometric Substitution
Find ∫1x2−2x−3  ⁣dx{\displaystyle\int}\frac{1}{\sqrt{x^2-2x-3}}\de{x}.
Practice: Complete the Square, Sec
Q:\bf{Q:}Evaluate ∫x2−4x+2 dx\displaystyle\int\sqrt{x^2-4x+2} \ dx
Practice: Use Sin
Q:\bf{Q:} Evaluate ∫x+29−x2dx\displaystyle\int\frac{x+2}{\sqrt{9-x^2}}dx
Trigonometric Substitution: Complete the Square, Sec
Evaluate ∫1x2x2−1 dx\displaystyle\int\frac{1}{x^2\sqrt{x^2-1}} \ dx
Trigonometric Substitution: Complete the Square, Tan
Evaluate ∫dxx2+4x+8\displaystyle\int\frac{dx}{\sqrt{x^2+4x+8}}