What is the integral from 0 to 0.5 of ln(1+x^4) ?

The integral from 0 to 0.5 of ln(1+x^4) is 0.00614…

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integral from 0 to 0.5 of ln(1+x^4)

\int_{0}^{0.5} ln (1 + x^{4}) d x

0.00614 \dots

Solution steps

\int_{0}^{0.5} ln (1 + x^{4}) d x

Apply Integration By Parts

: [x ln (1 + x^{4}) - \int \frac{4 x ^{4}}{1 + x ^{4}} d x]_{0}^{0.5}

= [x ln (1 + x^{4}) - \int \frac{4 x ^{4}}{1 + x ^{4}} d x]_{0}^{0.5}

\int \frac{4 x ^{4}}{1 + x ^{4}} d x = - \frac{1}{2} ln 2 x^{2} + 22 x + 2 - 2 arctan (2 x + 1) + \frac{1}{2} ln 2 x^{2} - 22 x + 2 - 2 arctan (2 x - 1) + 4 x

= [x ln (1 + x^{4}) - (- \frac{1}{2} ln 2 x^{2} + 22 x + 2 - 2 arctan (2 x + 1) + \frac{1}{2} ln 2 x^{2} - 22 x + 2 - 2 arctan (2 x - 1) + 4 x)]_{0}^{0.5}

Simplify

[x ln (1 + x^{4}) - (- \frac{1}{2} ln 2 x^{2} + 22 x + 2 - 2 arctan (2 x + 1) + \frac{1}{2} ln 2 x^{2} - 22 x + 2 - 2 arctan (2 x - 1) + 4 x)]_{0}^{0.5} : [x ln (1 + x^{4}) + 0.70710 \dots ln 2 x^{2} + 2.82842 \dots x + 2 + 1.41421 \dots arctan (1.41421 \dots x + 1) - 0.70710 \dots ln 2 x^{2} - 2.82842 \dots x + 2 + 1.41421 \dots arctan (1.41421 \dots x - 1) - 4 x]_{0}^{0.5}

= [x ln (1 + x^{4}) + 0.70710 \dots ln 2 x^{2} + 2.82842 \dots x + 2 + 1.41421 \dots arctan (1.41421 \dots x + 1) - 0.70710 \dots ln 2 x^{2} - 2.82842 \dots x + 2 + 1.41421 \dots arctan (1.41421 \dots x - 1) - 4 x]_{0}^{0.5}

Compute the boundaries:

0.00614 \dots

= 0.00614 \dots

\int_{1}^{2} (3 x^{2} + x) d x

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\int_{0}^{1} \frac{x}{2 x ^{2} + 2} d x

\int_{- 2}^{- 1} 2 x (7 - x^{2})^{3} d x

\int_{- 2}^{2} (4 x^{4} - 3 x^{2} + 5) d x

What is the integral from 0 to 0.5 of ln(1+x^4) ?
The integral from 0 to 0.5 of ln(1+x^4) is 0.00614…