Calc1.74

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\int {\frac  {x}{{\sqrt  {4+x^{2}}}}}\,dx\,

x=2\tan \theta \,

dx=2\sec ^{2}\theta \,d\theta \,

\int {\frac  {x}{{\sqrt  {4+x^{2}}}}}\,dx=\int {\frac  {2\tan \theta (2\sec ^{2}\theta )}{{\sqrt  {4+4\tan ^{2}\theta }}}}d\theta =2\int {\frac  {\tan \theta \sec ^{2}\theta }{{\sqrt  {\sec ^{2}\theta }}}}d\theta =2\int \tan \theta \sec \theta \,d\theta \,

Well, \sec \theta \tan \theta is just the derivative of \sec \theta so

\int {\frac  {x}{{\sqrt  {4+x^{2}}}}}\,dx=2\sec \theta +C\,

Now, we know that \tan \theta ={\frac  {x}{2}}. From that we can figure out that

\int {\frac  {x}{{\sqrt  {4+x^{2}}}}}\,dx=2\sec \theta +C=2{\frac  {{\sqrt  {x^{2}+4}}}{2}}+C={\sqrt  {x^{2}+4}}+C\,

Click here to see this problem done by substitution.


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