int(sin^(3)(x))/(cos^(2)(x))=

Rewrite Trig Functions: Rewrite the integral in terms of sine and cosine to make it easier to integrate. The integral is already in terms of sine and cosine, so no rewriting is necessary.Use Trig Identity: Recognize that sin^2(x) = 1 - cos^2(x) and rewrite the integral using this identity. int(sin^3(x))/(cos^2(x)) dx = int(sin(x) * (1 - cos^2(x)))/(cos^2(x)) dxExpand Integrand: Expand the integrand to separate the terms. int(sin(x) * (1 - cos^2(x)))/(cos^2(x)) dx = int(sin(x)/cos^2(x) dx) - int(sin(x) * cos^2(x)/cos^2(x) dx)Simplify Integral: Simplify the integral. int(sin(x)/cos^2(x) dx) - int(sin(x) * cos^2(x)/cos^2(x) dx) = int(sin(x)/cos^2(x) dx) - int(sin(x) dx)Recognize Integrals: Recognize that the first integral is the integral of tan(x) * sec(x) and the second integral is the integral of sin(x). int(sin(x)/cos^2(x) dx) - int(sin(x) dx) = int(tan(x) * sec(x) dx) - int(sin(x) dx)Integrate Separately: Integrate tan(x) * sec(x) and sin(x) separately. The integral of tan(x) * sec(x) is sec(x), and the integral of sin(x) is -cos(x). int(tan(x) * sec(x) dx) - int(sin(x) dx) = sec(x) - (-cos(x)) + CCombine Results: Combine the results to get the final answer. sec(x) - (-cos(x)) + C = sec(x) + cos(x) + C

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

$\int \frac{\sin ^{3}(x)}{\cos ^{2}(x)}=$

View step-by-step help

Home

Math Problems

Calculus

Evaluate definite integrals using the power rule

Full solution

\int \frac{\sin ^{3}(x)}{\cos ^{2}(x)}=

Rewrite Trig Functions: Rewrite the integral in terms of sine and cosine to make it easier to integrate. $\newline$ The integral is already in terms of sine and cosine, so no rewriting is necessary.
Use Trig Identity: Recognize that $\sin^2(x) = 1 - \cos^2(x)$ and rewrite the integral using this identity. $\newline$ $\int\frac{\sin^3(x)}{\cos^2(x)} \, dx = \int\frac{\sin(x) \cdot (1 - \cos^2(x))}{\cos^2(x)} \, dx$
Expand Integrand: Expand the integrand to separate the terms. $\newline$ $\int\frac{\sin(x) \cdot (1 - \cos^2(x))}{\cos^2(x)} dx = \int\frac{\sin(x)}{\cos^2(x)} dx$ - $\int\frac{\sin(x) \cdot \cos^2(x)}{\cos^2(x)} dx$
Simplify Integral: Simplify the integral. $\int(\frac{\sin(x)}{\cos^2(x)} dx) - \int(\sin(x) \cdot \frac{\cos^2(x)}{\cos^2(x)} dx) = \int(\frac{\sin(x)}{\cos^2(x)} dx) - \int(\sin(x) dx)$
Recognize Integrals: Recognize that the first integral is the integral of $\tan(x) \cdot \sec(x)$ and the second integral is the integral of $\sin(x)$ . $\int(\frac{\sin(x)}{\cos^2(x)} \, dx) - \int(\sin(x) \, dx) = \int(\tan(x) \cdot \sec(x) \, dx) - \int(\sin(x) \, dx)$
Integrate Separately: Integrate $\tan(x) \cdot \sec(x)$ and $\sin(x)$ separately. $\newline$ The integral of $\tan(x) \cdot \sec(x)$ is $\sec(x)$ , and the integral of $\sin(x)$ is $-\cos(x)$ . $\newline$ $\int(\tan(x) \cdot \sec(x) \, dx) - \int(\sin(x) \, dx) = \sec(x) - (-\cos(x)) + C$
Combine Results: Combine the results to get the final answer. $\sec(x) - (-\cos(x)) + C = \sec(x) + \cos(x) + C$