What is the integral from 1 to cos(x) of 8x^7 ?

The integral from 1 to cos(x) of 8x^7 is cos^8(x)-1

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integral from 1 to cos(x) of 8x^7

\int_{1}^{c o s (x)} 8 x^{7} d x

cos^{8} (x) - 1

Solution steps

\int_{1}^{c o s (x)} 8 x^{7} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 8 \cdot \int_{1}^{c o s (x)} x^{7} d x

Apply the Power Rule

= 8 [\frac{x ^{8}}{8}]_{1}^{c o s (x)}

Compute the boundaries:

\frac{cos ^{8} ( x )}{8} - \frac{1}{8}

= 8 (\frac{cos ^{8} ( x )}{8} - \frac{1}{8})

Simplify

= cos^{8} (x) - 1

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What is the integral from 1 to cos(x) of 8x^7 ?
The integral from 1 to cos(x) of 8x^7 is cos^8(x)-1