Question

Evaluate: $\int (14x^6-5x^4+7\sqrt{x}-2\cot x\csc x+5)dx$

Answer 1

Let $I=\int (14x^6-5x^4+7\sqrt{x}-2\cot x\csc x+5)dx$

$I=14\int x^{6}dx-5\int x^{4}dx+7\int x^{1/2}dx-2\int \cot x\text{cosec}xdx+5\int dx$

$I=14\times \frac{x^7}{7}-5\times \frac{x^5}{5}+7\times \frac{x^{3/2}}{3/2}+2\text{cosec}x+5x+C$

$I=2x^7-x^5+\frac{14}{3}x\sqrt{x}+2\text{cosec}x+5x+C$