integral of e+sec^2(θ)-e^θ

{ "query": { "display": "$$\\int\\:e+\\sec^{2}\\left(θ\\right)-e^{θ}dθ$$", "symbolab_question": "BIG_OPERATOR#\\int e+\\sec^{2}(θ)-e^{θ}dθ" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "eθ+\\tan(θ)-e^{θ}+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:e+\\sec^{2}\\left(θ\\right)-e^{θ}dθ=eθ+\\tan\\left(θ\\right)-e^{θ}+C$$", "input": "\\int\\:e+\\sec^{2}\\left(θ\\right)-e^{θ}dθ", "steps": [ { "type": "step", "primary": "Apply the Sum Rule: $$\\int{f\\left(x\\right){\\pm}g\\left(x\\right)}dx=\\int{f\\left(x\\right)}dx{\\pm}\\int{g\\left(x\\right)}dx$$", "result": "=\\int\\:edθ+\\int\\:\\sec^{2}\\left(θ\\right)dθ-\\int\\:e^{θ}dθ" }, { "type": "interim", "title": "$$\\int\\:edθ=eθ$$", "input": "\\int\\:edθ", "steps": [ { "type": "step", "primary": "Integral of a constant: $$\\int{a}dx=ax$$", "result": "=eθ" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:\\sec^{2}\\left(θ\\right)dθ=\\tan\\left(θ\\right)$$", "input": "\\int\\:\\sec^{2}\\left(θ\\right)dθ", "steps": [ { "type": "step", "primary": "Use the common integral: $$\\int\\:\\sec^{2}\\left(θ\\right)dθ=\\tan\\left(θ\\right)$$", "result": "=\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s79LMMseCPOF0sYSIRXMOQEqfMEKSvajjAtmuP2G8vRDuDa3kuA1QY9WtimNStJiaxMW1BmTDczzuQub1+8lCmJhkS3dlcCKpQTQcheuut7MkWHv1SxX3fydv4Mp2SaeNErYFVRedjMsXB1/Kk/9bb3W/Mg94S0N9we//Py6WzxN6" } }, { "type": "interim", "title": "$$\\int\\:e^{θ}dθ=e^{θ}$$", "input": "\\int\\:e^{θ}dθ", "steps": [ { "type": "step", "primary": "Use the common integral: $$\\int\\:e^{θ}dθ=e^{θ}$$", "result": "=e^{θ}" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7zB6jpd+QnOj73cBUx4/nPGYgdb5spS+z/MFHflaUY9WHJmlU1GZ0PuOxTUkXomMXMD75dWniGhYkJE6g4pDNkhmiNBsDw5B5GfZIoz6rep34zoIf5JblpCB6py3cqc63w==" } }, { "type": "step", "result": "=eθ+\\tan\\left(θ\\right)-e^{θ}" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=eθ+\\tan\\left(θ\\right)-e^{θ}+C", "meta": { "title": { "extension": "If $$\\frac{dF\\left(x\\right)}{dx}=f\\left(x\\right)$$ then $$\\int{f\\left(x\\right)}dx=F\\left(x\\right)+C$$" } } } ], "meta": { "solvingClass": "Integrals", "practiceLink": "/practice/integration-practice#area=main&subtopic=Sum%20Rule", "practiceTopic": "Integral Sum Rule" } }, "plot_output": { "meta": { "plotInfo": { "variable": "θ", "plotRequest": "y=eθ+\\tan(θ)-e^{θ}+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }