Calculus

Definite Integrals

Basic Properties of Integrals

         

If 01f(x)dx=4 \int_0^1 f(x) \, dx = 4 , what is the value of

012f(x)dx? \int_0^1 2 f(x) \, dx ?

If 01f(x)dx=2 \int_0^1 f(x) \, dx = 2 and 01g(x)dx=6 \int_0^1 g(x) \, dx = 6 , what is the value of

01(6f(x)+2g(x))dx? \int_0^1 \left( 6 f(x) + 2 g(x) \right) \, dx ?

If 04f(x)dx=3 \int_0 ^ {4} f(x) \, dx = 3 , what is the value of

04(3f(x))dx? \int_0^{4} \left( 3 - f(x) \right) \, dx ?

If 010f(x)dx=27 \int_0^{10} f(x) \, dx = 27 and 05f(x)dx=7 \int_0^5 f(x) \, dx = 7 , then what is the value of

510f(x)dx? \int_5^{10} f(x) \, dx ?

Suppose f(x)f(x) is an odd function and g(x)g(x) is an even function such that

05f(x)dx=7515f(x)dx=850g(x)dx=1515g(x)dx=2 \int_0 ^ {5} f(x) \, dx = 7 \hspace{.6cm} \int_{5}^{15} f(x) \, dx = 8 \\ \int_{-5}^{0} g(x) \, dx = 1 \hspace{.6cm} \int_{5}^{15} g(x) \, dx = 2

What is the value of 515(f(x)+g(x))dx? \int_{-5}^{15} \left( f(x) + g(x) \right) \, dx ?

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