Evaluate the integral using the given substitution.

Ask by Gibson Warner. in Malaysia

Mar 27,2025

Question

Evaluate the integral using the given substitution. $$ \int \frac{3}{x^{2}} \sin ^{2}\left(\frac{3}{x}\right) d x, u=-\frac{3}{x} $$

UpStudy AI · Accepted Answer

Calculate the integral $$ \int \frac{3}{x^{2}} \sin ^{2}\left(\frac{3}{x}ight) dx $$.
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int \frac{3}{x^{2}}	imes \sin^{2}\left(\frac{3}{x}ight) dx$$
- step1: Multiply the terms:
  $$\int \frac{3\sin^{2}\left(\frac{3}{x}ight)}{x^{2}} dx$$
- step2: Rewrite the expression:
  $$\int 3	imes \frac{\sin^{2}\left(\frac{3}{x}ight)}{x^{2}} dx$$
- step3: Use properties of integrals:
  $$3	imes \int \frac{\sin^{2}\left(\frac{3}{x}ight)}{x^{2}} dx$$
- step4: Use the substitution $$dx=-\frac{x^{2}}{3} dt$$ to transform the integral$$:$$
  $$3	imes \int \frac{\sin^{2}\left(\frac{3}{x}ight)}{x^{2}}	imes \left(-\frac{x^{2}}{3}ight) dt$$
- step5: Simplify:
  $$3	imes \int -\frac{\sin^{2}\left(\frac{3}{x}ight)}{3} dt$$
- step6: Use the substitution $$t=\frac{3}{x}$$ to transform the integral$$:$$
  $$3	imes \int -\frac{\sin^{2}\left(tight)}{3} dt$$
- step7: Simplify the expression:
  $$3	imes \int -\frac{1}{3}\sin^{2}\left(tight) dt$$
- step8: Use properties of integrals:
  $$3\left(-\frac{1}{3}ight)	imes \int \sin^{2}\left(tight) dt$$
- step9: Multiply the numbers:
  $$-\int \sin^{2}\left(tight) dt$$
- step10: Evaluate the integral:
  $$-\left(\frac{t}{2}-\frac{\sin\left(2tight)}{4}ight)$$
- step11: Simplify the expression:
  $$-\left(\frac{t}{2}-\frac{1}{4}\sin\left(2tight)ight)$$
- step12: Calculate:
  $$-\frac{t}{2}+\frac{1}{4}\sin\left(2tight)$$
- step13: Substitute back:
  $$-\frac{\frac{3}{x}}{2}+\frac{1}{4}\sin\left(2	imes \frac{3}{x}ight)$$
- step14: Multiply the terms:
  $$-\frac{\frac{3}{x}}{2}+\frac{1}{4}\sin\left(\frac{6}{x}ight)$$
- step15: Simplify the expression:
  $$-\frac{3}{2x}+\frac{1}{4}\sin\left(\frac{6}{x}ight)$$
- step16: Add the constant of integral C:
  $$-\frac{3}{2x}+\frac{1}{4}\sin\left(\frac{6}{x}ight) + C, C \in \mathbb{R}$$
The integral of $$ \frac{3}{x^{2}} \sin ^{2}\left(\frac{3}{x}ight) $$ with respect to $$ x $$ is $$ -\frac{3}{2x} + \frac{1}{4}\sin\left(\frac{6}{x}ight) + C $$, where $$ C $$ is an arbitrary constant.
The integral evaluates to $$ -\frac{3}{2x} + \frac{1}{4}\sin\left(\frac{6}{x}ight) + C $$.

Evaluate the integral using the given substitution.

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