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Mylo O'Moore

Answered question

2021-11-01

Convert the indefinite integral into definate integral using the interval [0,1]

\int (x^{5} + \frac{1}{5} x^{5} + \frac{1}{4} x^{4}) d x

coffentw

Skilled2021-11-02Added 103 answers

Step 1
Convert the indefinite integral to definite integral over [0,1] as,

\int (x^{5} + \frac{1}{5} x^{5} + \frac{1}{4} x^{4}) d x \Rightarrow \int_{0}^{1} (x^{5} + \frac{1}{5} x^{5} + \frac{1}{4} x^{4}) d x

Step 2
Compute integral as,

\int_{0}^{1} (x^{5} + \frac{1}{5} x^{5} + \frac{1}{4} x^{4}) d x = \int_{0}^{1} (\frac{6}{5} x^{5} + \frac{1}{4} x^{4}) d x

= \int_{0}^{1} \frac{6}{5} x^{5} d x + \int_{0}^{1} \frac{1}{4} x^{4} d x

= \frac{6}{5} {[\frac{x^{5 + 1}}{5 + 1}]}_{0}^{1} + \frac{1}{4} {[\frac{x^{4 + 1}}{4 + 1}]}_{0}^{1}

= \frac{1}{5} + \frac{1}{20}

= \frac{1}{4}

Step 3
Thus, the value of the integral is,

\int_{0}^{1} (x^{5} + \frac{1}{5} x^{5} + \frac{1}{4} x^{4}) d x = \frac{1}{4}

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