Additional Exercises in Integration

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Find indefinite integrals of the following functions of [math]x[/math].

Question i \[\int \! {\left(1+\frac{1}{x}\right)^{2}} \,\mathrm{d}x\] Solution
Question ii \[\int \! {x\left(3x^{2}+5\right)^{2}} \,\mathrm{d}x\] Solution
Question iii \[\int \! {2x \sqrt{x}} \,\mathrm{d}x\] Solution
Question iv \[\int \! {x \sqrt{2x^{2}+4}} \,\mathrm{d}x\]
Question v \[\int \! {\frac{x^{2}+1}{x^{4}\left(x^{3}+3\right)^{4}}} \,\mathrm{d}x\]
Question vi \[\int \! {\frac{1}{\sqrt{x}\left(\sqrt{x}+1\right)^{2}}} \,\mathrm{d}x\]
Question vii \[\int \! {\frac{\left(x^{\frac{2}{3}}+1\right)^{2}}{x^{\frac{1}{3}}}} \,\mathrm{d}x\]
Question viii \[\int \! {\sin{\left(2x-5\right)}} \,\mathrm{d}x\]
Question ix \[\int \! {\sec^{2}{4x}} \,\mathrm{d}x\]
Question x \[\int \! {\tan^{2}{x}} \,\mathrm{d}x\]
Question xi \[\int \! {\sin{x}\sin{2x}} \,\mathrm{d}x\]
Question xii \[\int \! {\frac{\sin{x}}{\cos^{3}{x}}} \,\mathrm{d}x\]
Question xiii \[\int \! {\sin^{2}{x}} \,\mathrm{d}x\]
Question xiv \[\int \! {\tan^{4}{x}} \,\mathrm{d}x\]
Question xv \[\int \! {\frac{1}{4+x^{2}}} \,\mathrm{d}x\]
Question xvi \[\int \! {\frac{1}{\sqrt{4-x^{2}}}} \,\mathrm{d}x\]
Question xvii \[\int \! {\frac{1}{\sqrt{12-3x^{2}}}} \,\mathrm{d}x\]
Question xviii \[\int \! {\sqrt{4-9x^{2}}} \,\mathrm{d}x\]
Question xix \[\int \! {xe^{x^{2}+1}} \,\mathrm{d}x\]
Question xx \[\int \! {2^{4x}} \,\mathrm{d}x\]
Question xxi \[\int \! {x \cdot e^{x^{2}}} \,\mathrm{d}x\]
Question xxii \[\int \! {\frac{1}{10^{3x}}} \,\mathrm{d}x\]
Question xxiii \[\int \! {\frac{1}{2x-3}} \,\mathrm{d}x\]
Question xxiv \[\int \! {\frac{x}{2x^{2}-4}} \,\mathrm{d}x\]
Question xxv \[\int \! {\frac{x^{3}+1}{x\left(x^{3}+4\right)}} \,\mathrm{d}x\]
Question xxvi \[\int \! {\frac{\sec^{2}{x}}{\tan{x}}} \,\mathrm{d}x\]
Question xxvii \[\int \! {\cos^{2}{x}\cot{x}} \,\mathrm{d}x\]
Question xxviii \[\int \! {\tan^{3}{x}} \,\mathrm{d}x\]
Question xxix \[\int \! {\frac{\sec^{2}{x}}{\sqrt{1+2\tan{x}}}} \,\mathrm{d}x\]
Question xxx \[\int \! {\frac{1}{x\left(x+1\right)}} \,\mathrm{d}x\]
Question xxxi \[\int \! {\frac{x-2}{x^{2}-1}} \,\mathrm{d}x\]
Question xxxii \[\int \! {\frac{3x+4}{\left(x+2\right)^{2}}} \,\mathrm{d}x\]
Question xxxiii \[\int \! {\frac{1}{x^{2}+2x+5}} \,\mathrm{d}x\]
Question xxxiv \[\int \! {\frac{a-x}{a^{2}+x^{2}}} \,\mathrm{d}x\]
Question xxxv \[\int \! {\frac{2\sqrt{2}}{x^{2}+2x-1}} \,\mathrm{d}x\]
Question xxxvi \[\int \! {\frac{1}{4x^{2}+8x+13}} \,\mathrm{d}x\]
Question xxxvii \[\int \! {\frac{1}{\sqrt{2x-x^{2}}}} \,\mathrm{d}x\]
Question xxxviii \[\int \! {\frac{18x}{9x^{2}+6x+4}} \,\mathrm{d}x\]
Question xxxix \[\int \! {\frac{x^{2}-x+1}{x^{2}\left(x-1\right)}} \,\mathrm{d}x\]
Question xxxx \[\int \! {\frac{\sin{x}}{1-4\cos^{2}{x}}} \,\mathrm{d}x\]
Question xxxxi \[\int \! {\frac{1}{\cos^{2}{\left(\frac{1}{2}x\right)}-\sin^{2}{\left(\frac{1}{2}x\right)}}} \,\mathrm{d}x\]
Question xxxxii \[\int \! {3x \cos{3x}} \,\mathrm{d}x\]
Question xxxxiii \[\int \! {x \sec^{2}{2x}} \,\mathrm{d}x\]
Question xxxxiv \[\int \! {\ln{\sqrt{x}}} \,\mathrm{d}x\]
Question xxxxv \[\int \! {x^{2} e^{-x}} \,\mathrm{d}x\]
Question xxxxvi \[\int \! {x^{3} \ln{x}} \,\mathrm{d}x\]