Revision on Integration

Give two real life applications of integration.

Find $$\int_{ }^{ }\left(8x^3+6x^{\frac{1}{2}}-5\right)\ dx$$   giving each term in its simplest form.

Evaluate $$\int_0^{\frac{\pi}{6}}\cos x\ \cos3x\ dx$$   .

The curve C has equation $$y=f\left(x\right),\ x>0$$  , where $$\frac{\text{d}y}{\text{d}x}=3x\ -\frac{5}{\sqrt[]{x}}-2$$  . Given that the point P (4, 5) lies on C, find f(x). 

The anti-derivative of $$\sin x$$  is

(a)  

The diagram shows the curve with equation $$y=2-\frac{1}{\sqrt[]{x}}$$  . The shaded region bounded by the curve, the x-axis and the lines x = 1 and x = 4 is rotated through 360° about the x-axis to form the solid S. Find the volume of the container correct to 3 significant figures.

Find $$\int_{ }^{ }x\ \ln\ x$$  dx.

$$f\left(x\right)=\frac{5x+1}{\left(1-x\right)\left(1+2x\right)}$$  , find $$\int_0^{\frac{1}{2}}f\left(x\right)dx$$  .

Shade the area representing $$\int_2^4x^2-6x+8\ dx$$  .

Integrate $$x^e+e^x+e^e$$  .