What is the integral from 0 to 1/7 of arcsin(7x) ?

The integral from 0 to 1/7 of arcsin(7x) is pi/(14)-1/7

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integral from 0 to 1/7 of arcsin(7x)

\int_{0}^{\frac{1}{7}} arcsin (7 x) d x

\frac{π}{14} - \frac{1}{7}

Decimal Notation

0.08154 \dots

Solution steps

\int_{0}^{\frac{1}{7}} arcsin (7 x) d x

Apply Integration By Parts

: [x arcsin (7 x) - \int \frac{7 x}{1 - 49 x ^{2}} d x]_{0}^{\frac{1}{7}}

= [x arcsin (7 x) - \int \frac{7 x}{1 - 49 x ^{2}} d x]_{0}^{\frac{1}{7}}

\int \frac{7 x}{1 - 49 x ^{2}} d x = - \frac{1}{7} 1 - 49 x^{2}

= [x arcsin (7 x) - (- \frac{1}{7} 1 - 49 x^{2})]_{0}^{\frac{1}{7}}

Simplify

[x arcsin (7 x) - (- \frac{1}{7} 1 - 49 x^{2})]_{0}^{\frac{1}{7}} : [\frac{1}{7} (7 x arcsin (7 x) + 1 - 49 x^{2})]_{0}^{\frac{1}{7}}

= [\frac{1}{7} (7 x arcsin (7 x) + 1 - 49 x^{2})]_{0}^{\frac{1}{7}}

Compute the boundaries:

\frac{π}{14} - \frac{1}{7}

= \frac{π}{14} - \frac{1}{7}

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What is the integral from 0 to 1/7 of arcsin(7x) ?
The integral from 0 to 1/7 of arcsin(7x) is pi/(14)-1/7