Evaluate the integral.

{eq}\int_{0}^{\frac{\pi}{2}} \sin^3 \theta \cos^5 \theta \, \mathrm{d} \theta {/eq}


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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Learn more about this topic:

Definite Integrals: Definition


Chapter 12 / Lesson 6

A 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.

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