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Integration Questions

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Question 1


Find the primitive of:

a) $2x$  a s v d
b) $3x^2$  a s v d
c) $7x^6$  a s v d
d) $\frac32x^{\frac12}$  a s v d
e) $1.2x^{0.2}$  a s v d
f) $-x^{-2}$  a s v d


Question 2


Evaluate:

a) $\int 4x^3\,dx$  a s v d
b) $\int 9x^8\,dx$  a s v d
c) $\int -2x^{-3}\,dx$  a s v d
d) $\int \frac12x^{-\frac12}\,dx$  a s v d


Question 3


Evaluate:

a) $\int x\,dx$  a s v d
b) $\int x^2\,dx$  a s v d
c) $\int x^5\,dx$  a s v d
d) $\int x^{-2}\,dx$  a s v d
e) $\int x^{1.5}\,dx$  a s v d
f) $\int x^{\frac43}\,dx$  a s v d
g) $\int x^{\frac12}\,dx$  a s v d
h) $\int x^{\frac25}\,dx$  a s v d


Question 4


Evaluate:

a) $\int 8x\,dx$  a s v d
b) $\int 3x\,dx$  a s v d
c) $\int 6x^2\,dx$  a s v d
d) $\int 4x^5\,dx$  a s v d
e) $\int 9x^{17}\,dx$  a s v d
f) $\int -2x^3\,dx$  a s v d
g) $\int -8x^{\frac12}\,dx$  a s v d
h) $\int -7x^{\frac34}\,dx$  a s v d
i) $\int \frac12x^7\,dx$  a s v d
j) $\int \frac65x^2\,dx$  a s v d
k) $\int -\frac12x^-3\,dx$  a s v d
l) $\int -\frac97x^-5\,dx$  a s v d


Question 5


Evaluate. Leave answers as improper fractions where necessary.

a) $\int_{1}^{2} x\,dx$  a s v d
b) $\int_{2}^{3} x^2\,dx$  a s v d
c) $\int_{2}^{6} 4x^3\,dx$  a s v d
d) $\int_{0}^{2} 2x^7\,dx$  a s v d
e) $\int_{1}^{4} x^{\frac12}\,dx$  a s v d
f) $\int_{3}^{5} x^{-2}\,dx$  a s v d
g) $\int_{-1}^{1} \frac45x^{9}\,dx$  a s v d
h) $\int_{-2}^{-1} -\frac32x^{-3}\,dx$  a s v d


Question 6


Simplify and then evaluate:

a) $\int x\times x\,dx$  a s v d
b) $\int (2x)^2\,dx$  a s v d
c) $\int 4\times x\times x^2\,dx$  a s v d
d) $\int \frac{1}{x^2}\,dx$  a s v d
e) $\int \frac{1}{x^4}\,dx$  a s v d
f) $\int \frac{2}{x^2}\,dx$  a s v d
g) $\int \frac{3}{4x^5}\,dx$  a s v d
h) $\int \frac{4}{x^8}\times\frac{2x^3}{5}\,dx$  a s v d


Question 7


Evaluate. Leave answers as fractions where necessary.

a) $\int_{3}^{5} \frac{1}{x^2}\,dx$  a s v d
b) $\int_{1}^{2} 3(2x)^2\,dx$  a s v d
c) $\int_{-2}^{-1} \frac{3}{x^7}\,dx$  a s v d
d) $\int_{\frac14}^{\frac12} \left(\frac{4}{x^2}\right)\times\frac1x\,dx$  a s v d


Question 8


Evaluate:

a) $\int \left(2x+3x^2\right)\,dx$  a s v d
b) $\int \left(x+x^{-3}\right)\,dx$  a s v d
c) $\int \left(3x^5+2x^{\frac12}\right)\,dx$  a s v d
d) $\int \left(\frac{1}{x^3}+8x^{0.2}\right)\,dx$  a s v d
e) $\int \left(3x(1+4x)\right)\,dx$  a s v d
f) $\int \left(x+2\right)^2\,dx$  a s v d
g) $\int 3\left(x+\frac1x\right)^2\,dx$  a s v d
h) $\int \left(2x\left(2x+1\right)^2+\frac1x\times\frac4x\right)\,dx$  a s v d


Question 9


Evaluate. Leave answers as improper fractions where necessary.

a) $\int_{0}^{\frac12} \left(x^5+16x\right)\,dx$  a s v d
b) $\int_{1}^{2} \left(3x^2-x^{-2}\right)\,dx$  a s v d
c) $\int_{-1}^{0} \left(3-x\right)^2\,dx$  a s v d
d) $\int_{1}^{6} 2\left(\frac3x+x^2\right)^2\,dx$  a s v d


Question 10


Evaluate:

a) $\int \frac{2x+x^2}{x}\,dx$  a s v d
b) $\int \frac{x+9}{\sqrt x}\,dx$  a s v d
c) $\int \frac{(2x^2+5x)^2}{3x}\,dx$  a s v d


Question 11


Find the value of $a$ given that:

a) $\int_{a}^{2} x^3\,dx=\frac{15}{4}$  a s v d
b) $\int_{0}^{a} (3x-2)^2\,dx=245$  a s v d
c) $\int_{4}^{9} a\sqrt{x}\,dx=\frac{76}{3}$  a s v d