Comprehensive assessment covering integration by substitution, standard integrals giving inverse trigonometric functions, integration by parts, partial fractions, trigonometric substitutions, and the t-substitution.
$\displaystyle\int \dfrac{1}{1 + x^2}\,dx$ equals:
$\displaystyle\int \dfrac{1}{\sqrt{1 - x^2}}\,dx$ equals:
The integration by parts formula is:
$\displaystyle\int \dfrac{1}{a^2 + x^2}\,dx$ equals:
To evaluate $\displaystyle\int \dfrac{x + 1}{(x - 1)(x + 2)}\,dx$, the first step is to:
Using the substitution $u = x^2$, $\displaystyle\int 2x\,e^{x^2}\,dx$ equals:
$\displaystyle\int \dfrac{1}{x}\,dx$ equals:
Using integration by parts, $\displaystyle\int x\cos x\,dx$ equals:
The substitution $x = a\sin\theta$ is most useful for integrals containing:
$\displaystyle\int \dfrac{1}{\sqrt{4 - x^2}}\,dx$ equals:
Under the t-substitution $t = \tan\dfrac{x}{2}$, $\sin x$ equals:
Under the t-substitution $t = \tan\dfrac{x}{2}$, $dx$ equals:
$\displaystyle\int_0^1 \dfrac{1}{1 + x^2}\,dx$ equals:
$\displaystyle\int \sec^2 x\,dx$ equals:
In the partial fraction decomposition $\dfrac{1}{(x - 1)(x + 1)} = \dfrac{A}{x - 1} + \dfrac{B}{x + 1}$, the value of $A$ is:
1. Find $\displaystyle\int (2x + 3)^5\,dx$. (2 marks)
2. Evaluate $\displaystyle\int \dfrac{1}{9 + x^2}\,dx$. (2 marks)
3. Use integration by parts to find $\displaystyle\int x e^x\,dx$. (2 marks)
4. Express $\dfrac{5x - 1}{(x - 1)(x + 2)}$ in partial fractions. (3 marks)
5. Find $\displaystyle\int \dfrac{1}{\sqrt{25 - x^2}}\,dx$. (2 marks)
6. Evaluate $\displaystyle\int_0^{1/2} \dfrac{1}{\sqrt{1 - x^2}}\,dx$. (2 marks)
7. Use the substitution $u = x^2 + 1$ to find $\displaystyle\int \dfrac{x}{x^2 + 1}\,dx$. (2 marks)
8. Find $\displaystyle\int \ln x\,dx$. (3 marks)
9. Use integration by parts to find $\displaystyle\int x \sin x\,dx$. (2 marks)
10. Find $\displaystyle\int \tan^2 x\,dx$. (2 marks)