To determine

To evaluate:

The given definite integral ∫011+7x 3dx.

Answer

4528

Explanation

1) Concept:

i) The substitution rule for definite integral:

If g'(x) is a continuous function on a,b whose f is continuous on range of u=g(x), then ∫abfgxg'(x)dx=∫g(a)g(b)f(u)du. Here,g(x) is substituted as u and then the differentiation is g’(x)dx =du.

ii) Indefinite integral

∫xn dx=xn+1n+1+C   (n≠-1)

iii) ∫abcfxdx=c∫abfxdx

2) Given:

∫011+7x 3dx

3) Calculation:

The given integral is

∫011+7x 3dx

Here, use the substitution method.

Substitute 1+7x=u  so   x=u-17

Differentiating with respect to x,

dx=17du

The limits change, and the new limits of integration are calculated by substituting

for x=0, u=1+70=1 &  for x=1, u=1+71=8

Therefore, the given integral becomes

∫011+7x 3dx=∫18u 317du

=17∫18u1/3du

Integrating this, we get

=17u1/3+113+118

=17u4/343-12

=328(84/3-14/3)

=328(16-1)

=32815

=4528

Conclusion:

Therefore,

∫011+7x 3dx=4528