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Well, I'm an AP Calculus BC student, and I always liked to get ahead, so before the year started, I went through Calc I and II. Now that I'm in the class with a 100+, I'm simply refreshing myself on the Calc II portion. My brother decided to give me a seemingly simple problem that turned out more complicated than I expected. Any assistance would be greatly appreciated.

Problem: Evaluate integral

[tex]\int \sin{x} \sqrt{(\cos{x})^2 + 1}dx[/tex]

I decided to use substitution initially, setting u = cos x. dx = du/-sin(x), thus eliminating the sin(x) and leaving [tex]\int \sqrt{u^2 + 1}du [/tex]

I then thought my best bet would be to approach this with trignometric substitution. I said tan([tex]\theta[/tex]) = u so

[tex]\sec{\theta}^2d\theta = du [/tex]

[tex]\int (\sec{\theta})^2\sqrt{(\tan(\theta)} +1)dx [/tex]

I then evaluated [tex]\sqrt{ (\tan{\theta})^2 +1 } = (\sec{\theta})[/tex]

Thus, my integral was simplified (relatively) down to

[tex] \int (\sec{\theta})^3 d\theta [/tex]

It just gets worse from there. Am I making some horrid mistake in my trignometric substitution or before or have I completely approached this incorrectly? I have more work that I've done, but I want to ensure that this is correct so far. Any help would be great. Thanks

Problem: Evaluate integral

[tex]\int \sin{x} \sqrt{(\cos{x})^2 + 1}dx[/tex]

I decided to use substitution initially, setting u = cos x. dx = du/-sin(x), thus eliminating the sin(x) and leaving [tex]\int \sqrt{u^2 + 1}du [/tex]

I then thought my best bet would be to approach this with trignometric substitution. I said tan([tex]\theta[/tex]) = u so

[tex]\sec{\theta}^2d\theta = du [/tex]

[tex]\int (\sec{\theta})^2\sqrt{(\tan(\theta)} +1)dx [/tex]

I then evaluated [tex]\sqrt{ (\tan{\theta})^2 +1 } = (\sec{\theta})[/tex]

Thus, my integral was simplified (relatively) down to

[tex] \int (\sec{\theta})^3 d\theta [/tex]

It just gets worse from there. Am I making some horrid mistake in my trignometric substitution or before or have I completely approached this incorrectly? I have more work that I've done, but I want to ensure that this is correct so far. Any help would be great. Thanks

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