alculate the integral and express your answer in
implest form.

Ask by Romero Paul. in the United States

Jan 21,2025

Question

alculate the integral and express your answer in implest form. $$ \int_{-1}^{2} \frac{15}{\sqrt[4]{5 x+6}} \mathrm{~d} x $$

UpStudy AI · Accepted Answer

Calculate the integral $$ \int_{-1}^{2} \frac{15}{\sqrt[4]{5x+6}} dx $$.
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int_{-1}^{2} \frac{15}{\sqrt[4]{5x+6}} dx$$
- step1: Evaluate the power:
  $$\int_{-1}^{2} \frac{15}{\left(5x+6ight)^{\frac{1}{4}}} dx$$
- step2: Evaluate the integral:
  $$\int \frac{15}{\left(5x+6ight)^{\frac{1}{4}}} dx$$
- step3: Rewrite the expression:
  $$\int 15	imes \frac{1}{\left(5x+6ight)^{\frac{1}{4}}} dx$$
- step4: Use properties of integrals:
  $$15	imes \int \frac{1}{\left(5x+6ight)^{\frac{1}{4}}} dx$$
- step5: Use the substitution $$dx=\frac{1}{5} dt$$ to transform the integral$$:$$
  $$15	imes \int \frac{1}{\left(5x+6ight)^{\frac{1}{4}}}	imes \frac{1}{5} dt$$
- step6: Simplify:
  $$15	imes \int \frac{1}{5\left(5x+6ight)^{\frac{1}{4}}} dt$$
- step7: Use the substitution $$t=5x+6$$ to transform the integral$$:$$
  $$15	imes \int \frac{1}{5t^{\frac{1}{4}}} dt$$
- step8: Rewrite the expression:
  $$15	imes \int \frac{1}{5}	imes \frac{1}{t^{\frac{1}{4}}} dt$$
- step9: Use properties of integrals:
  $$15	imes \frac{1}{5}	imes \int \frac{1}{t^{\frac{1}{4}}} dt$$
- step10: Multiply the numbers:
  $$3	imes \int \frac{1}{t^{\frac{1}{4}}} dt$$
- step11: Evaluate the integral:
  $$3	imes \frac{t^{-\frac{1}{4}+1}}{-\frac{1}{4}+1}$$
- step12: Simplify:
  $$3	imes \frac{4}{3}t^{\frac{3}{4}}$$
- step13: Multiply the numbers:
  $$4t^{\frac{3}{4}}$$
- step14: Substitute back:
  $$4\left(5x+6ight)^{\frac{3}{4}}$$
- step15: Return the limits:
  $$\left(4\left(5x+6ight)^{\frac{3}{4}}ight)\bigg |_{-1}^{2}$$
- step16: Calculate the value:
  $$28$$
The integral of $$ \frac{15}{\sqrt[4]{5x+6}} $$ from -1 to 2 is 28.
The integral evaluates to 28.

alculate the integral and express your answer in
implest form.

Upstudy AI Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Calculus Questions