int_(0)^((pi)/(4))sin^(3)(x)cos(x)dx

Given Integral: We are given the integral to evaluate: int_(0)^((pi)/(4))sin^(3)(x)cos(x)dx To solve this integral, we can use the power-reducing formula for sin^2(x) or we can use a substitution method. Since we have a sin^3(x) term and a cos(x) term, it's convenient to use substitution. Let's let u = sin(x), which means du = cos(x)dx.Substitution Method: Now we need to change the limits of integration to match our substitution. When x = 0, u = sin(0) = 0. When x = pi/4, u = sin(pi/4) = sqrt(2)/2. So our new limits of integration are from u = 0 to u = sqrt(2)/2.Limits of Integration: Substituting u for sin(x) and du for cos(x)dx, we get: int_(0)^(sqrt(2)/2)u^3du This is a simple power integral that we can evaluate using the power rule for integrals.Power Rule Integration: Using the power rule for integrals, we integrate u^3 with respect to u: int u^3 du = (u^4)/4 + CApply Limits: Now we apply the limits of integration to the antiderivative: ((sqrt(2)/2)^4)/4 - (0^4)/4Calculate Value: Calculating the value of the antiderivative at the limits, we get: ((sqrt(2)/2)^4)/4 = ((2^(1/2))^4)/2^2 = (2^2)/2^2 = 1/4 And since (0^4)/4 = 0, we don't need to subtract anything.Final Result: So the value of the integral is: 1/4 - 0 = 1/4

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

$\int_{0}^{\frac{\pi}{4}} \sin ^{3}(x) \cos (x) d x$

View step-by-step help

Home

Math Problems

Calculus

Evaluate definite integrals using the chain rule

Full solution

\int_{0}^{\frac{\pi}{4}} \sin ^{3}(x) \cos (x) d x

Given Integral: We are given the integral to evaluate: $\newline$ $\int_{0}^{\frac{\pi}{4}}\sin^{3}(x)\cos(x)dx$ $\newline$ To solve this integral, we can use the power-reducing formula for $\sin^2(x)$ or we can use a substitution method. Since we have a $\sin^3(x)$ term and a $\cos(x)$ term, it's convenient to use substitution. Let's let $u = \sin(x)$ , which means $du = \cos(x)dx$ .
Substitution Method: Now we need to change the limits of integration to match our substitution. When $x = 0$ , $u = \sin(0) = 0$ . When $x = \frac{\pi}{4}$ , $u = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$ . So our new limits of integration are from $u = 0$ to $u = \frac{\sqrt{2}}{2}$ .
Limits of Integration: Substituting $u$ for $ext{sin}(x)$ and $du$ for $ext{cos}(x)dx$ , we get: $\newline$ ext{int}_{0}^{rac{ ext{sqrt}(2)}{2}}u^3du $\newline$ This is a simple power integral that we can evaluate using the power rule for integrals.
Power Rule Integration: Using the power rule for integrals, we integrate $u^3$ with respect to $u$ : $\int u^3 \, du = \frac{u^4}{4} + C$
Apply Limits: Now we apply the limits of integration to the antiderivative: $\left(\frac{\sqrt{2}}{2}\right)^4/4 - \left(0^4\right)/4$
Calculate Value: Calculating the value of the antiderivative at the limits, we get: $\newline$ $(\frac{\sqrt{2}}{2})^4$ / $4$ = $(2^{\frac{1}{2}})^4$ / $2$ ^ $2$ = \frac{ $2$ ^ $2$ }{ $2$ ^ $2$ } = \frac{ $1$ }{ $4$ }\) $\newline$ And since $(0^4)$ / $4$ = $0$ , we don't need to subtract anything.
Final Result: So the value of the integral is: $\frac{1}{4} - 0 = \frac{1}{4}$