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7-2.Definite Integral question bank (Mathematics)

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Q1.

If $f(x)$ is a function satisfying $f^{\prime}(x)=f(x)$ with $f(0)=1$ and $g(x)$ is a function that satisfies $f(x)+g(x)=x^2$. Then the value of the integral $\int_0^1 f(x) g(x) d x$ is

(A)

$e-\frac{e^2}{2}-\frac{5}{2}$

(B)

$e+\frac{e^2}{2}-\frac{3}{2}$

(C)

$e-\frac{e^2}{2}-\frac{3}{2}$

(D)

$e+\frac{e^2}{2}+\frac{5}{2}$

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Q2.

The value of $\int_{0}^{1} x^{2}(1-x^{2})^{3/2} dx$ is

(A)

$\frac{1}{32}$

(B)

$\frac{\pi}{8}$

(C)

$\frac{\pi}{16}$

(D)

$\frac{\pi}{32}$

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