Answer to Question #234648 in Calculus for Sayem

Question

Find the derivative of the function.

1. Integrate[(Sqrt[t+Sin[t]]),{t,x,1}]
2. Integrate[Sin[Power[t,4]],{t,2x,3x+4}]

Answer 1

1.

$F(z)=\int^z_0 \sqrt{t+sint}dt$

$F'(z)= \sqrt{z+sinz}$

$\int^1_x \sqrt{t+sint}dt=\int^1_0 \sqrt{t+sint}dt-\int^x_0 \sqrt{t+sint}dt=F(1)-F(x)$

$\frac{d}{dx}(F(1)-F(x))=F'(1)-F'(x)=\sqrt{1+sin1}-\sqrt{x+sinx}$

2.

$F(z)=\int^z_0 sin^4 tdt$

$F'(z)= sin^4 z$

$\int^{3x+4}_{2x} sin^4 tdt=\int^{3x+4}_{0} sin^4 tdt-\int^{2x}_{0} sin^4 tdt=F(3x+4)-F(2x)$

$\frac{d}{dx}(F(3x+4)-F(2x))=3F'(3x+4)-2F'(2x)=$

$=3sin^4(3x+4)-2sin^4(2x)$