What is the integral from 1 to cos(x) of 5t^4 ?

The integral from 1 to cos(x) of 5t^4 is cos^5(x)-1

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integral from 1 to cos(x) of 5t^4

\int_{1}^{c o s (x)} 5 t^{4} d t

cos^{5} (x) - 1

Solution steps

\int_{1}^{c o s (x)} 5 t^{4} d t

Take the constant out:

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

= 5 \cdot \int_{1}^{c o s (x)} t^{4} d t

Apply the Power Rule

: [\frac{t ^{5}}{5}]_{1}^{c o s (x)}

= 5 [\frac{t ^{5}}{5}]_{1}^{c o s (x)}

Compute the boundaries:

\frac{cos ^{5} ( x )}{5} - \frac{1}{5}

= 5 (\frac{cos ^{5} ( x )}{5} - \frac{1}{5})

Simplify

= cos^{5} (x) - 1

\int_{1}^{4} \frac{6}{x} d x

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\int_{0}^{x} e^{x^{3}} d x

\int_{0}^{4} x + x d x

What is the integral from 1 to cos(x) of 5t^4 ?
The integral from 1 to cos(x) of 5t^4 is cos^5(x)-1