Question

Evaluate the definite integrals.
${\int }_0^{\frac{\pi }{4}}(2se{c}^{2}x+x^3+2)dx$

Answer 1

${\int }_0^{\frac{\pi }{4}}(2se{c}^{2}x+x^3+2)dx=2{\int }_0^{\frac{\pi }{4}}se{c}^{2}xdx+{\int }_0^{\frac{\pi }{4}}{x}^{3}dx+2{\int }_0^{\frac{\pi }{4}}1dx=2[tanx{]}_0^{\frac{\pi }{4}}+\frac{1}{4}[x^4{]}_0^{\frac{\pi }{4}}+2[x{]}_0^{\frac{\pi }{4}}=2tan\frac{\pi }{4}+\frac{1}{4}(\frac{\pi }{4}{)}^4+2\left(\frac{\pi }{4}\right)=2+\frac{\pi }{2}+\frac{{\pi }^4}{1024}$