Evaluate the integral.
{eq}\int_{0}^{\frac{\pi}{2}} \sin^3 \theta \cos^5 \theta \, \mathrm{d} \theta {/eq}
Question:
Evaluate the integral.
{eq}\int_{0}^{\frac{\pi}{2}} \sin^3 \theta \cos^5 \theta \, \mathrm{d} \theta {/eq}
Definite Integral:
An integral of the form {eq}\displaystyle \int_a^b f(t)\,dt {/eq} which can be expressed as the difference between the values of the integral at specified upper and lower limits of the independent variable.
We use definite integral property here:
{eq}\int\limits_a^b {f(x)dx = - } \int\limits_b^a {f(x)dx} {/eq},
We also use the following formula here:
{eq}\begin{align} \int {t^ndt} &= t^{n+1} + c \end{align} {/eq},
where c is constant of integration.
Answer and Explanation: 1
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{eq}\begin{align} I &= \int_{0}^{\frac{\pi}{2}} \sin^3 \theta \cos^5 \theta \, \mathrm{d} \theta \\ &= \int\limits_0^{\pi /2} {\sin \theta...
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Chapter 12 / Lesson 6A definite integral is found as the limit between a line graphed from an equation, and the x-axis, either positive or negative. Learn how this limit is identified in practical examples of definite integrals.