MATHEMATICS II

where c=c₁+c₂+c₃.  After these calculations we obtain the integral as

The original integral becomes

and v=2x³ Inserting above equalities into ∫udv=uv-∫vdu we get;

where c is an arbitrary real number.

where c is an arbitrary real number. 

is a function such that its derivative is 

We can take

since

Substituting these values into the integral we get 

where c is an arbitrary real number.

To use the integration by parts method, let u=7x+6 and dv=e^-2xdx. Then we have du=7dx and 

Inserting above equalities into ∫udv=uv-∫vdu  we get;

where c is an arbitrary real number.

where c is an arbitrary real number. For F (-1) =4 we have F(-1) =3/8+c=4 which gives c=29/8.

where c is an arbitrary real number.

for any arbitrary real number c.

.

Substituting these values into ∫udv=uv-∫vdu  we get;

for any arbitrary real number c.