Question

1.Integrate 1/(1+x^2) for limit (0,1)
2.what is the value of integer 8x^3dx

Answer 1

$\bf \large1.$

$\int ^{1} _{0} \frac{1}{1 + {x}^{2} } dx$

$\sf \purple{ we \: know \: that : }$

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

$\implies [\tan ^{ - 1}x ]^{1} _{0}$

$\implies \tan ^{ - 1}(1) - \tan ^{ - 1}(0)$

$\implies \frac{\pi}{4} - 0$

$\orange{ \therefore \frac{\pi}{4} }$

$\bf \large2.$

$\int8 {x}^{3} dx$

$\implies 8 \int {x}^{3} dx$

$\sf \purple{ we \: know \: that : }$

$\blue{ \boxed{\int {x}^{n} dx = \frac{ {x}^{n + 1} }{n + 1} +I _{ c} }}$

$\implies( 8 ) ( \frac{ {x}^{3 + 1} }{3 + 1} ) + I _{ c}$

$\implies( 8 ) ( \frac{ {x}^{4} }{4} ) + I _{ c}$

$\orange{\therefore2 {x}^{4} + I _{ c}}$

HOPE IT HELPS!!