To determine

To evaluate:

The given indefinite integral ∫x2x+58dx.

Answer

140(2x+5)10-536(2x+50)9+C

Explanation

1) Concept:

i) The substitution rule:

If u=g(x) is a differentiable function whose range is I and f is continuous on I, then ∫f(gx)g'xdx=∫f(u)du.

ii) Indefinite integral

∫xn dx=xn+1n+1+C  n≠-1

iii)

∫ab[fx+gx] dx=∫abfxdx+∫abgxdx

iv)

∫abcfxdx=c∫abfxdx

2) Given:

∫x2x+58dx

3) Calculation:

The given integral is

∫x2x+58dx

Here, use the substitution method.

Thus, substitute 2x+5=u    so  x=u-52 therefore   dx=du2,

Therefore, the given integral becomes

∫x2x+58dx

=∫u-52(u)8⁡du2

=∫(u9-5u8)2⁡du2

=14∫(u9-5u8)  ⁡du

=14∫u9du-54∫u8du

By using concept ii),

=14u9+19+1-54u8+18+1+C

=14u1010-54u99+C

=u1040-5u936+C

Resubstitute 2x+5=u.

Hence, the solution becomes

=140(2x+5)10-536(2x+50)9+C

Conclusion:

Therefore,

∫x2x+58dx=140(2x+5)10-536(2x+50)9+C