Evaluate the integral using the given substitution.

Ask by Christensen Fitzgerald. in Malaysia

Mar 27,2025

Question

Evaluate the integral using the given substitution. $$ \int 6\left(y^{6}+5 y^{3}+5\right)^{3}\left(2 y^{5}+5 y^{2}\right) d y, u=y^{6}+5 y^{3}+5 $$

UpStudy AI · Accepted Answer

Calculate the integral $$ \int 6\left(y^{6}+5 y^{3}+5ight)^{3}\left(2 y^{5}+5 y^{2}ight) dy $$.
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int 6\left(y^{6}+5y^{3}+5ight)^{3}\left(2y^{5}+5y^{2}ight) dy$$
- step1: Multiply the terms:
  $$\int \left(12y^{5}+30y^{2}ight)\left(y^{6}+5y^{3}+5ight)^{3} dy$$
- step2: Use the substitution $$dy=\frac{1}{6y^{5}+15y^{2}} dt$$ to transform the integral$$:$$
  $$\int \left(12y^{5}+30y^{2}ight)\left(y^{6}+5y^{3}+5ight)^{3}	imes \frac{1}{6y^{5}+15y^{2}} dt$$
- step3: Simplify:
  $$\int 2\left(y^{6}+5y^{3}+5ight)^{3} dt$$
- step4: Use the substitution $$t=y^{6}+5y^{3}+5$$ to transform the integral$$:$$
  $$\int 2t^{3} dt$$
- step5: Use properties of integrals:
  $$2	imes \int t^{3} dt$$
- step6: Evaluate the integral:
  $$2	imes \frac{t^{3+1}}{3+1}$$
- step7: Simplify:
  $$2	imes \frac{t^{4}}{4}$$
- step8: Reduce the fraction:
  $$1	imes \frac{t^{4}}{2}$$
- step9: Multiply the terms:
  $$\frac{t^{4}}{2}$$
- step10: Substitute back:
  $$\frac{\left(y^{6}+5y^{3}+5ight)^{4}}{2}$$
- step11: Add the constant of integral C:
  $$\frac{\left(y^{6}+5y^{3}+5ight)^{4}}{2} + C, C \in \mathbb{R}$$
The integral of $$6\left(y^{6}+5 y^{3}+5ight)^{3}\left(2 y^{5}+5 y^{2}ight)$$ with respect to $$y$$ is $$\frac{(y^{6}+5y^{3}+5)^{4}}{2} + C$$, where $$C$$ is an arbitrary constant.
The integral evaluates to $$\frac{(y^{6}+5y^{3}+5)^{4}}{2} + C$$.

Evaluate the integral using the given substitution.

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