Definite integration：練習問題

Evaluate the definite integral: $$ \int_1^2 \left(x^2 + \frac{1}{x}\right) dx $$

Evaluate the definite integral:
$$\int_0^1 x \sqrt{1-x^2} \, dx$$

Evaluate the definite integral:
\(\int_0^1 x \tan^{-1} x dx\)

Evaluate the definite integral:
\(\int_{-\pi/4}^{\pi/4} (x \cos x + \cos^2 x) dx\)

Evaluate the definite integral:
\( \int_0^{\pi} \frac{x \sin x}{3 + \sin^2 x} dx \)

Calculate the value of \(\int_{1}^{2} e^x dx\).

Find the exact area of the region bounded by the curve $$y = \frac{1}{(2x+1)^2}$$, the x-axis, and the lines $$x=0$$ and $$x=1$$.

Evaluate the definite integral: $$\int_{-2}^{2} |x^2 - 1| \, dx$$

Evaluate the definite integral \( \int_{0}^{\frac{\pi}{4}} (\sec^2 x + e^{2x}) \, dx \). Give your answer in terms of \( e \) and \( \pi \).

Consider the integral \( J = \int_{0}^{1} x(1-x)^5 \, dx \).
(a) By using the substitution \( u = 1-x \), or otherwise, show that \( J = \int_{0}^{1} (u^5 - u^6) \, du \).
(b) Hence, find the exact value of \( J \).