Mathematics

Integral of $$\displaystyle \frac{(4x^{2}-2\sqrt{x})}{x} +\frac{1}{1+x^{2}}-5 co\sec ^{2}x$$ is 


SOLUTION
Let $$\displaystyle I=\int { \left( \frac { (4x^{ 2 }-2\sqrt { x } ) }{ x } +\frac { 1 }{ 1+x^{ 2 } } -5co\sec ^{ 2 } x \right) dx } \\ ={ I }_{ 1 }+{ I }_{ 2 }+{ I }_{ 3 }$$

Where $$\displaystyle { I }_{ 1 }=\int { \frac { (4x^{ 2 }-2\sqrt { x } ) }{ x } dx } $$

Put $$\displaystyle t=\sqrt { x } \Rightarrow dt=\frac { 1 }{ 2\sqrt { x }  } dx$$

$${ I }_{ 1 }=4\int { \left( 2{ t }^{ 3 }-1 \right)  } dt=2{ t }^{ 4 }-4t=2{ x }^{ 2 }-4\sqrt { x } $$

$$\displaystyle { I }_{ 2 }=\int { \frac { 1 }{ 1+x^{ 2 } } dx } =\tan ^{ -1 }{ x } $$

$${ I }_{ 3 }=-5\int { co\sec ^{ 2 } x } dx=5\cot { x } $$
Therefore
$$I=2{ x }^{ 2 }-4\sqrt { x } +\tan ^{ -1 }{ x } +5\cot { x } $$
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Subjective Medium Published on 17th 09, 2020
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