8.1 The Fundamental Theorem of Calculus and standard integrals — Practise

Work through these on your own. For each indefinite integral, add the constant of integration, then check by differentiating your answer.

The questions and the answers below come from a set that is checked by computer every time the book is built.

Find $\displaystyle\int x^5\,dx$ .
Find $\displaystyle\int 3x^2\,dx$ .
Find $\displaystyle\int \left(4x^3 - 2x\right)dx$ .
Find $\displaystyle\int \frac{1}{x^2}\,dx$ .
Find $\displaystyle\int \sqrt{x}\,dx$ .
Find $\displaystyle\int \frac{1}{x}\,dx$ for $x > 0$ .
Find $\displaystyle\int e^{3x}\,dx$ .
Find $\displaystyle\int \sin 2x\,dx$ .
Find $\displaystyle\int \cos 4x\,dx$ .
Find $\displaystyle\int \left(e^{2x} + \sin x\right)dx$ .
The gradient of a curve is $\dfrac{dy}{dx} = 3x^2$ , and the curve passes through $(1, 5)$ . Find the equation of the curve.
Use the Fundamental Theorem of Calculus to evaluate $\displaystyle\int_0^2 3x^2\,dx$ .

Answers and concise solutions

1. The integral

Answer: Raise the power by one and divide by the new power: $\displaystyle\int x^5\,dx = \dfrac{x^6}{6} + C$ .

2. The integral

Answer: The constant multiple stays outside: $3 \times \dfrac{x^3}{3} + C = x^3 + C$ .

3. The integral

Answer: Integrate term by term: $4 \times \dfrac{x^4}{4} - 2 \times \dfrac{x^2}{2} + C = x^4 - x^2 + C$ .

4. The integral

Answer: Write $\dfrac{1}{x^2} = x^{-2}$ . Then $\displaystyle\int x^{-2}\,dx = \dfrac{x^{-1}}{-1} + C = -\dfrac{1}{x} + C$ .

5. The integral

Answer: Write $\sqrt{x} = x^{1/2}$ . Then $\displaystyle\int x^{1/2}\,dx = \dfrac{x^{3/2}}{3/2} + C = \dfrac{2}{3}x^{3/2} + C$ .

6. The integral

Answer: The power rule does not cover $n = -1$ , so use the standard result $\displaystyle\int \dfrac{1}{x}\,dx = \ln x + C$ .

7. The integral

Answer: The factor is $\dfrac{1}{k} = \dfrac{1}{3}$ , so $\displaystyle\int e^{3x}\,dx = \dfrac{1}{3}e^{3x} + C$ .

8. The integral

Answer: The integral of sine is minus cosine, with the factor $\dfrac{1}{2}$ : $\displaystyle\int \sin 2x\,dx = -\dfrac{1}{2}\cos 2x + C$ .

9. The integral

Answer: The integral of cosine is sine, with the factor $\dfrac{1}{4}$ : $\displaystyle\int \cos 4x\,dx = \dfrac{1}{4}\sin 4x + C$ .

10. The integral

Answer: Integrate each term: $\dfrac{1}{2}e^{2x} - \cos x + C$ . The sign of the cosine term changes because the integral of sine is minus cosine.

11. The equation of the curve

Answer: Integrating gives $y = x^3 + C$ . Substituting the point $(1, 5)$ gives $5 = 1 + C$ , so $C = 4$ and $y = x^3 + 4$ .

12. The value of the integral

Answer: An antiderivative is $x^3$ , so $\displaystyle\int_0^2 3x^2\,dx = \Big[x^3\Big]_0^2 = 8 - 0 = 8$ .