Find the constant of integration, C, if: y = \int 12x(x^{2} + 3)^{5} dx and the curve passes...

Question:

Find the constant of integration, {eq}C {/eq}, if:

{eq}\displaystyle\; y = \int 12x\left(x^{2} + 3\right)^{5}\, dx {/eq}

and the curve passes through the point {eq}(c,d) {/eq}.

{eq}(c = 6, \quad d = 9) {/eq}

Indefinite Integral:

We have been given an indefinite integral that has integrand as polynomials. We have to find the constant of integration. We need to evaluate the integral first. There are many methods to evaluate integrals like the substitution method.

Answer and Explanation:

$$\displaystyle\; y = \int 12x\left(x^{2} + 3\right)^{5}\, dx $$

We will do the following substitution:

$$x^2+3=t\\ 2xdx=dt\\ y=\int 6t^5dt $$

Now we will apply the standard integration formula:

$$y=\frac{6t^6}{6}+C\\ y=t^6+C $$

Where C is the constant of integration whose value we will find by the given condition:

$$(6,9)\\ 9=6^6+C\\ C=-46647 $$


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