Check the solution to "If f is continuous and integral 0 to..."

Nannie Mack

Answered question

2021-10-12

If f is continuous and integral 0 to 9 f(x)dx=4, find integral 0 to 3

x f (x^{2}) d x

opsadnojD

Skilled2021-10-13Added 95 answers

\int_{0}^{3} x f (x^{2}) d x

Substitute

x^{=} u

and

2 x d x = d u

= \frac{1}{2} \int_{0}^{3} f (x^{2}) (2 x d x)

Limits of Integration will change from

\int_{0}^{3}

{\int_{0}^{2}}^{{{(3)}^{2}}} = \int_{0}^{9}

= \frac{1}{2} \int_{0}^{9} f (u) d u

Remember that

\int_{a}^{b} f (u) d u = \int_{a}^{b} f (x) d x

= \frac{1}{2} \int_{0}^{9} f (x) d x = 4

= \frac{1}{2} \times 4 = 2

Result:

\int_{0}^{3} x f (x^{2}) d x = 2

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