26. $$ \int_{-\frac{\pi}{4}}^{\frac{\pi}{2}} \cos \left(2 t+\frac{\pi}{4}\right) d t $$

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UpStudy AI · Accepted Answer

Calculate or simplify the expression $$ \int_{-\frac{\pi}{4}}^{\frac{\pi}{2}} \cos(2t+\frac{\pi}{4}) dt $$.
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int_{-\frac{\pi }{4}}^{\frac{\pi }{2}} \cos\left(2t+\frac{\pi }{4}ight) dt$$
- step1: Evaluate the integral:
  $$\int \cos\left(2t+\frac{\pi }{4}ight) dt$$
- step2: Use the substitution $$dt=\frac{1}{2} dv$$ to transform the integral$$:$$
  $$\int \cos\left(2t+\frac{\pi }{4}ight)	imes \frac{1}{2} dv$$
- step3: Simplify:
  $$\int \frac{1}{2}\cos\left(2t+\frac{\pi }{4}ight) dv$$
- step4: Use the substitution $$v=2t+\frac{\pi }{4}$$ to transform the integral$$:$$
  $$\int \frac{1}{2}\cos\left(vight) dv$$
- step5: Use properties of integrals:
  $$\frac{1}{2}	imes \int \cos\left(vight) dv$$
- step6: Evaluate the integral:
  $$\frac{1}{2}\sin\left(vight)$$
- step7: Substitute back:
  $$\frac{1}{2}\sin\left(2t+\frac{\pi }{4}ight)$$
- step8: Return the limits:
  $$\left(\frac{1}{2}\sin\left(2t+\frac{\pi }{4}ight)ight)\bigg |_{-\frac{\pi }{4}}^{\frac{\pi }{2}}$$
- step9: Calculate the value:
  $$0$$
La valeur de l'intégrale est de 0.
L'intégrale vaut 0.

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