Exercises XVII. (See p. 263 for the Answers.)

(1) Find ${\displaystyle \int y\,dx}$ when ${\displaystyle y^{2}=4ax}$.

(2) Find ${\displaystyle \int {\frac {3}{x^{4}}}\,dx}$.

(3) Find ${\displaystyle \int {\frac {1}{a}}x^{3}\,dx}$.

(4) Find ${\displaystyle \int (x^{2}+a)\,dx}$.

(5) Find ${\displaystyle 5x^{-{\frac {7}{2}}}}$.

(6) Find ${\displaystyle \int (4x^{3}+3x^{2}+2x+1)\,dx}$.

(7) Find ${\displaystyle {\dfrac {dy}{dx}}={\dfrac {ax}{2}}+{\dfrac {bx^{2}}{3}}+{\dfrac {cx^{3}}{4}}}$; find ${\displaystyle y}$.

(8) Find ${\displaystyle \int \left({\frac {x^{2}+a}{x+a}}\right)dx}$.

(9) Find ${\displaystyle \int (x+3)^{3}\,dx}$.

(10) Find ${\displaystyle \int (x+2)(x-a)\,dx}$.

(11) Find ${\displaystyle \int ({\sqrt {x}}+{\sqrt[{3}]{x}})3a^{2}\,dx}$.

(12) Find ${\displaystyle \int (\sin \theta -{\tfrac {1}{2}})\,{\frac {d\theta }{3}}}$.

(13) Find ${\displaystyle \int \cos ^{2}a\theta \,d\theta }$.

(14) Find ${\displaystyle \int \sin ^{2}\theta \,d\theta }$.

(15) Find ${\displaystyle \int \sin ^{2}a\theta \,d\theta }$.

(16) Find ${\displaystyle \int \epsilon ^{3x}\,dx}$.

(17) Find ${\displaystyle \int {\dfrac {dx}{1+x}}}$.

(18) Find ${\displaystyle \int {\dfrac {dx}{1-x}}}$.