Asksia AI LOGO


Posted 3 months ago
I2=0π2sin4xdxI_{2}=\int_{0}^{\frac{\pi}{2}} \sin ^{4} x d x
Sign in to unlock the answer
Answer from Sia
Posted 3 months ago
Solution by Steps
step 1
To find the integral of sin4(x) \sin^4(x) from 0 0 to π2 \frac{\pi}{2} , we use the power-reduction formula for even powers of sine
step 2
The integral of sin4(x) \sin^4(x) can be expressed as 0π23814cos(2x)+132cos(4x)dx \int_0^{\frac{\pi}{2}} \frac{3}{8} - \frac{1}{4} \cos(2x) + \frac{1}{32} \cos(4x) \, dx
step 3
Integrating term by term, we get [3x818sin(2x)+1128sin(4x)]0π2 \left[\frac{3x}{8} - \frac{1}{8} \sin(2x) + \frac{1}{128} \sin(4x)\right]_0^{\frac{\pi}{2}}
step 4
Evaluating the antiderivative at the bounds 0 0 and π2 \frac{\pi}{2} , we find (3π160+0)(00+0) \left(\frac{3\pi}{16} - 0 + 0\right) - \left(0 - 0 + 0\right)
step 5
The result of the definite integral is 3π16 \frac{3\pi}{16}
3π16 \frac{3\pi}{16}
Key Concept
Power-Reduction Formula for Even Powers of Sine
The integral of sin4(x) \sin^4(x) from 0 0 to π2 \frac{\pi}{2} can be computed by using the power-reduction formula to rewrite the integral in terms of cosine, which simplifies the integration process.

Not the question you are looking for? Ask here!

Enter question by text

Enter question by image

Upgrade to Asksia Pro

Join a AskSia's Pro Plan, and get 24/7 AI tutoring for your reviews, assignments, quizzes and exam preps.

Unlimited chat query usages
Strong algorithms that better know you
Early access to new release features
Study Other Question