(1+t^3)^(1/2) −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 −3 -2.5 −2 -1.5 −1 -0.5 0 0.5 1 1.5 2 2.5 3 x y
Exercise∫ 1 2 x ( 1 + t 3 ) d t \int_1^{2x}\left(\sqrt{1+t^3}\right)dt ∫ 1 2 x ( 1 + t 3 ) d t
Step-by-step Solution
1
The integral ∫ 1 + t 3 d t \int\sqrt{1+t^3}dt ∫ 1 + t 3 d t
[ 2 5 t 3 + 1 ( t 4 + − 1 6 ( 3 ) 3 4 − 1 6 ( t + 1 ) t 2 − t + 1 F ( arcsin ( − ( − 1 ) 5 6 ( t + 1 ) 3 4 ) ∣ − 1 3 ) + t ) ] 1 2 x \left[\frac{2}{5\sqrt{t^3+1}}\left(t^4+\sqrt[6]{-1}\sqrt[4]{\left(3\right)^{3}}\sqrt{\sqrt[6]{-1}\left(t+1\right)}\sqrt{t^2-t+1}F\left(\arcsin\left(\frac{\sqrt{-\sqrt[6]{\left(-1\right)^{5}}\left(t+1\right)}}{\sqrt[4]{3}}\right)\Big\vert \sqrt[3]{-1}\right)+t\right)\right]_{1}^{2x} ⎣ ⎡ 5 t 3 + 1 2 ⎝ ⎛ t 4 + 6 − 1 4 ( 3 ) 3 6 − 1 ( t + 1 ) t 2 − t + 1 F ⎝ ⎛ arcsin ⎝ ⎛ 4 3 − 6 ( − 1 ) 5 ( t + 1 ) ⎠ ⎞ ∣ ∣ 3 − 1 ⎠ ⎞ + t ⎠ ⎞ ⎦ ⎤ 1 2 x
Final answer to the exercise [ 2 5 t 3 + 1 ( t 4 + − 1 6 ( 3 ) 3 4 − 1 6 ( t + 1 ) t 2 − t + 1 F ( arcsin ( − ( − 1 ) 5 6 ( t + 1 ) 3 4 ) ∣ − 1 3 ) + t ) ] 1 2 x \left[\frac{2}{5\sqrt{t^3+1}}\left(t^4+\sqrt[6]{-1}\sqrt[4]{\left(3\right)^{3}}\sqrt{\sqrt[6]{-1}\left(t+1\right)}\sqrt{t^2-t+1}F\left(\arcsin\left(\frac{\sqrt{-\sqrt[6]{\left(-1\right)^{5}}\left(t+1\right)}}{\sqrt[4]{3}}\right)\Big\vert \sqrt[3]{-1}\right)+t\right)\right]_{1}^{2x} ⎣ ⎡ 5 t 3 + 1 2 ⎝ ⎛ t 4 + 6 − 1 4 ( 3 ) 3 6 − 1 ( t + 1 ) t 2 − t + 1 F ⎝ ⎛ arcsin ⎝ ⎛ 4 3 − 6 ( − 1 ) 5 ( t + 1 ) ⎠ ⎞ ∣ ∣ 3 − 1 ⎠ ⎞ + t ⎠ ⎞ ⎦ ⎤ 1 2 x