Free Printable Differentiation & Integration

1.

Given $y=5x^3+6x^2-3x+2$ . Find $\frac{dy}{dx}$ .

a)

$30x^2+12$

b)

$15x^2+12x-3$

c)

$15x^2-12x+3$

d)

$15x^3+12x^2-3x$

2.

Given $f\left(x\right)=\left(1+\sqrt[]{x}\right)^2$ . Find $\frac{dy}{dx}$ .

a)

$\frac{1}{\left(\sqrt[]{x}\right)^3+1}$

b)

$x^{\frac{3}{2}}+1$

c)

$2\left(1+\sqrt[]{x}\right)$

d)

$\frac{1}{\sqrt[]{x}}+1$

3.

Find the first derivatives of $y=\left(x^2+3\right)\left(x^2-2x\right)^4$ .

a)

$\left(x^2-2x\right)^3\left(4x^3-2x^2+2x-6\right)$

b)

$\left(x^2-2x\right)^3\left(10x^3-12x^2+24x-24\right)$

c)

$\left(x^2-2x\right)^4\left(10x^3-12x^2+24x-24\right)$

d)

$\left(x^2-2x\right)^3\left(4\left(x^2-2x\right)\left(2x-2\right)+\left(x^2-2x\right)\right)$

4.

Find $\frac{dy}{dx}$ for the functio, $f\left(x\right)=\frac{\operatorname{cosec}\left(x+1\right)}{e^{2x}}$ .

a)

$\frac{1}{e^{4x}}\left(-\operatorname{cosec}\left(x+1\right)\cot\left(x+1\right)\right)-\left(\operatorname{cosec}\left(x+1\right)e^{\left(x\right)}\right)$

b)

$-\frac{\operatorname{cosec}\left(x+1\right)\left[e^{2x}\cot\left(x+1\right)+2e^{2x}\right]}{e^{2x}}$

c)

$e^x\left(-\operatorname{cosec}\left(x+1\right)\cot\left(x+1\right)\right)-\left(\operatorname{cosec}\left(x+1\right)e^{\left(x\right)}\right)$

d)

$-\frac{\operatorname{cosec}\left(x+1\right)\left[e^{2x}\cot\left(x+1\right)+2e^{2x}\right]}{e^{4x}}$

5.

Find $\frac{dy}{dx}$ for $y=\tan^4\left(2x+1\right)$ .

a)

$8\tan^4\left(2x+1\right)\sec^2\left(2x+1\right)$

b)

$4\tan^3\left(2x+1\right)\sec^2\left(2x+1\right)$

c)

$4\tan^3\left(2x+1\right)\sec^2\left(2x+1\right)$

d)

$8\tan^3\left(2x+1\right)\sec^2\left(2x+1\right)$

6.

Find $\frac{dy}{dx}$ for $y=3x^2\left(5x+1\right)$ by using product rule.

a)

$45x^2-6x$

b)

$45x^2+6x^{-1}$

c)

$45x+6$

d)

$45x^2+6x$

7.

Find $\frac{dy}{dx}$ for $y=\frac{3x+2^{ }}{2x+1}$ by using quotient rule.

a)

$-\frac{1}{\left(2x+1\right)^2}$

b)

$\frac{1}{\left(2x+1\right)^2}$

c)

$\left(2x+1\right)^{-2}$

d)

$2x+1$

8.

∫(3x⁴+2)⁴(12x³)dx

a)

½(3x⁴+2)⁴+C

b)

(3x⁴+2)⁴+C

c)

⅕(3x⁴+2)⁵+C

d)

⅓(3x⁴+2)⁶+C

9.

$\int\left(6\sqrt{x}-1\right)dx$

a)

$6x^{\frac{3}{2}}+x+C$

b)

$4x^{\frac{3}{2}}-x+C$

c)

$3x^{-\frac{1}{2}}-x+C$

d)

I didn't look at my notes to see how to do this one.

10.

Solve $\int_{ }\left(2-3\tan x\right)^4\sec^2x\ dx$

a)

$\frac{1}{5}\left(2-3\tan x\ \right)^5+C$

b)

$-\frac{1}{3}\left(2-3\tan x\ \right)^5+C$

c)

$-\frac{1}{5}\left(2-3\tan x\ \right)^5+C$

d)

$-\frac{1}{15}\left(2-3\tan x\ \right)^5+C$

11.

Which of the following is TRUE ?

a)

$\int_{ }^{ }\sin\ x\ \ dx\ =\ \cos\ x\ +C$

b)

$\int_{ }^{ }e^x\ \ dx\ =\ \frac{e^{x+1}}{x+1}\ +\ C$

c)

$\int_{ }^{ }\ 2x\ \ dx\ =\ x^2\ +C$

d)

$\int_{ }^{ }\ \frac{1}{x^2}\ \ dx\ =\ \ln\left|x^2\right|+C$

12.

For evaluating $\int_0^2x\left(8x^2-6\right)^7dx$ , the most suitable method is to use

a)

Integration by substitution

b)

Integration by partial fraction

c)

Integration by parts

d)

Indefinite integrals

13.

To integrate $\int_{ }^{ }xe^{x^2}dx$ , the most suitable method is to use

a)

Integration by substitution

b)

Integration by partial fraction

c)

Integration by parts

d)

Indefinite integrals

14.

To solve

\int\frac{1-2x}{x^2+2x+1}dx

, the most suitable method is to use

a)

Integration by substitution

b)

Integration by parts

c)

Integration by partial fraction

d)

Definite Integral

15.

Use substitution to evaluate the integral $\int\left(3x^3+5\right)^5\cdot27x^2dx$

a)

$\frac{1}{2}\left(3x^3+5\right)^6+C$

b)

$\frac{2}{5}\left(3x^3+5\right)^5+C$

c)

$\frac{5}{6}\left(3x^3+5\right)^6+C$

d)

$\frac{3}{5}\left(3x^3+5\right)^5+C$

Differentiation & Integration

Total questions: 15

Worksheet time: 30mins

Similar Resources on Wayground