∫(arctanx)²*1/(1+x²) dx
=∫(arctanx)² d(arctanx)
=(1/3)(arctanx)³+C
25+9x²=5²+(3x)²,令3x=5tanu,dx=(5/3)sec²u du
25+9x²=25+25tan²u=25sec²u
∴∫dx/(25+9x²)=(5/3)(1/25)∫sec²u/sec²u du
=(1/15)∫du=(1/15)u+C
=(1/15)arctan(3x/5)+C
∫(arctanx)²*1/(1+x²) dx
=∫(arctanx)² d(arctanx)
=(1/3)(arctanx)³+C
25+9x²=5²+(3x)²,令3x=5tanu,dx=(5/3)sec²u du
25+9x²=25+25tan²u=25sec²u
∴∫dx/(25+9x²)=(5/3)(1/25)∫sec²u/sec²u du
=(1/15)∫du=(1/15)u+C
=(1/15)arctan(3x/5)+C