三角函数求积分技巧 三角函数求积分
\u4e09\u89d2\u51fd\u6570\u7684\u5b9a\u79ef\u5206\u516c\u5f0f\u222bsin ²x dx =1/2x -1/4 sin 2x + C
\u222b cos ²x dx = 1/2+1/4 sin 2x + C
\u222b tan²x dx =tanx -x+ C
\u222b cot ²x dx =-cot x-x+ C
\u222b sec ²x dx =tanx + C
\u222b csc ²x dx =-cot x+ C
\u222barcsin x dx = xarcsin x+\u221a\uff08bai1-x²\uff09+C
\u222barccosx dx = xarccos x-\u221a\uff081-x²\uff09+C
\u222barctan x dx = xarctan x-1/2ln\uff081+x²\uff09+C
\u222barc cot x dx =xarccot x+1/2ln\uff081+x²\uff09+C
\u222barcsec xdx =xarcsec x-ln\u2502dux+\u221a\uff08x²-1\uff09\u2502+C
\u222barccsc x dx =xarccsc x+ln\u2502x+\u221a\uff08x²-1\uff09\u2502+C
\u6269\u5c55\u8d44\u6599\uff1a
\u5982\u679c\u4e00\u4e2a\u51fd\u6570f\u5728\u67d0\u4e2a\u533a\u95f4\u4e0a\u9ece\u66fc\u53ef\u79ef\uff0c\u5e76\u4e14\u5728\u6b64\u533a\u95f4\u4e0a\u5927\u4e8e\u7b49\u4e8e\u96f6\u3002\u90a3\u4e48\u5b83\u5728\u8fd9\u4e2a\u533a\u95f4\u4e0a\u7684\u79ef\u5206\u4e5f\u5927\u4e8e\u7b49\u4e8e\u96f6\u3002\u5982\u679cf\u52d2\u8d1d\u683c\u53ef\u79ef\u5e76\u4e14\u51e0\u4e4e\u603b\u662f\u5927\u4e8e\u7b49\u4e8e\u96f6\uff0c\u90a3\u4e48\u5b83\u7684\u52d2\u8d1d\u683c\u79ef\u5206\u4e5f\u5927\u4e8e\u7b49\u4e8e\u96f6\u3002
\u4f5c\u4e3a\u63a8\u8bba\uff0c\u5982\u679c\u4e24\u4e2a\u4e0a\u7684\u53ef\u79ef\u51fd\u6570f\u548cg\u76f8\u6bd4\uff0cf\uff08\u51e0\u4e4e\uff09\u603b\u662f\u5c0f\u4e8e\u7b49\u4e8eg\uff0c\u90a3\u4e48f\u7684\uff08\u52d2\u8d1d\u683c\uff09\u79ef\u5206\u4e5f\u5c0f\u4e8e\u7b49\u4e8eg\u7684\uff08\u52d2\u8d1d\u683c\uff09\u79ef\u5206\u3002
\u5982\u679c\u9ece\u66fc\u53ef\u79ef\u7684\u975e\u8d1f\u51fd\u6570f\u5728\u4e0a\u7684\u79ef\u5206\u7b49\u4e8e0\uff0c\u90a3\u4e48\u9664\u4e86\u6709\u9650\u4e2a\u70b9\u4ee5\u5916\uff0cf = 0\u3002\u5982\u679c\u52d2\u8d1d\u683c\u53ef\u79ef\u7684\u975e\u8d1f\u51fd\u6570f\u5728\u4e0a\u7684\u79ef\u5206\u7b49\u4e8e0\uff0c\u90a3\u4e48f\u51e0\u4e4e\u5904\u5904\u4e3a0\u3002\u5982\u679c\u4e2d\u5143\u7d20A\u7684\u6d4b\u5ea6\u03bc \uff08A\uff09\u7b49\u4e8e0\uff0c\u90a3\u4e48\u4efb\u4f55\u53ef\u79ef\u51fd\u6570\u5728A\u4e0a\u7684\u79ef\u5206\u7b49\u4e8e0\u3002
\u5982\u56fe\u6240\u793a\uff0c\u8fd9\u662f\u7531\u5bf9\u79f0\u6027\u51b3\u5b9a\u7684
f(x)=[sin(x)]^4\u7684\u5468\u671f\u662f\u03c0\uff0c\u5bf9\u79f0\u8f74\u662fx=k\u03c0/2\uff08k\u4e3a\u6574\u6570\uff09\u3002\u7531\u5bf9\u79f0\u6027\u3001\u5b9a\u79ef\u5206\u7684\u51e0\u4f55\u6027\u8d28\u77e5\u539f\u5f0f\u6210\u7acb
(sinx)^2=(1-cos2x)/2\uff0c\u56e0\u6b64(sinx)^2\u7684\u5468\u671f\u4e0ecos2x\u76f8\u540c\uff0c\u7b49\u4e8e\u03c0
(sinx)^4=[(sinx)^2]^2=[(1-cos2x)/2]^2=(1-cos2x)^2/4=[1-2cos2x+(cos2x)^2]/4=[1-2cos2x+(1+cos4x)/2]/4\uff0c(sinx)^4\u7684\u5468\u671f\u662fcos2x\u7684\u5468\u671f\uff08\u7b49\u4e8e\u03c0\uff09\u548ccos4x\u7684\u5468\u671f\uff08\u7b49\u4e8e\u03c0/2\uff09\u7684\u6700\u5c0f\u516c\u500d\u6570\uff0c\u6545(sinx)^4\u7684\u5468\u671f\u662f\u03c0
\u4ee5\u6b64\u7c7b\u63a8\uff0c(sinx)^(2k)=a + b*cos2x + c*cos4x + d*cos6x + ...\uff08k=1,2,3...\uff09\uff0c\u5468\u671f\u662f\u03c0\u3001\u03c0/2\u3001\u03c0/3\u2026\u2026\u7684\u6700\u5c0f\u516c\u500d\u6570\uff0c\u5373(sinx)^(2k)\u7684\u5468\u671f\u662f\u03c0
\u800c(sinx)^(2k)\u7684\u5bf9\u79f0\u8f74\u662fx=k\u03c0/2\uff08k\u4e3a\u6574\u6570\uff09\uff0c\u5373\u5728[0,\u03c0]\u5185\u7684\u56fe\u5f62\u5173\u4e8ex=\u03c0/2\u5bf9\u79f0\uff0c\u6545\u6709\u222b(0\u2192\u03c0/2)(sinx)^(2k)dx=\u222b(\u03c0/2\u2192\u03c0)(sinx)^(2k)dx=(1/2)\u222b(0\u2192\u03c0)(sinx)^(2k)dx
\u7531\u6b64\u63a8\u51fa\u222b(0\u21922\u03c0)(sinx)^4*dx=2\u222b(0\u2192\u03c0)(sinx)^4*dx=2*2\u222b(0\u2192\u03c0/2)(sinx)^4*dx=4\u222b(0\u2192\u03c0/2)(sinx)^4*dx
绛旓細鏂规硶涓 1銆佸ぇ澶氭暟澶氶」寮忛傜敤鐨勭Н鍒嗗叕寮忋傛瘮濡傚椤瑰紡锛歽 = a*x^n.銆2銆佺郴鏁伴櫎浠(n+1)锛岀劧鍚庢寚鏁板姞涓1銆傛崲鍙ヨ瘽璇磞 = a*x^n 鐨勭Н鍒嗘槸y = (a/n+1)*x^(n+1).銆3銆佸浜庝笉瀹氱Н鍒嗭紝涓涓椤瑰紡瀵瑰簲澶氫釜锛屾墍浠ヨ鍔犱笂绉垎甯告暟C銆傚洜姝ゆ湰渚嬬殑鏈缁堢粨鏋滄槸y = (a/n+1)*x^(n+1) + C銆 ...
绛旓細瑙ｇ瓟鏂规硶濡備笅锛鈭玸in^2xdx =鈭1/2-cos2x/2dx =x/2-sin2x/4+C cos2x =1-2sin^2x sin^2x =(1-cos2x)/2 =1/2-cos2x/2銆備竴浜涚畝鍗曠殑鍚湁涓夎鍑芥暟鐨勭Н鍒嗭紝鍙湪涓夎鍑芥暟绉垎琛ㄤ腑鎵惧埌銆傝屼笁瑙掔Н鍒嗘槸涓绉嶉潪鍒濈瓑鍑芥暟锛屽惈鏈変笁瑙掑嚱鏁扮殑涓绉嶇Н鍒嗐備竴浜涚畝鍗曠殑鍚湁涓夎鍑芥暟鐨勭Н鍒嗭紝鍙湪涓夎鍑芥暟绉垎...
绛旓細1銆佸皢tan⁴x闄嶉樁锛屽彲杩愮敤涓夎鍑芥暟鐨鍩烘湰鍏崇郴sec²x=tan²x+1杩涜鍖栫畝 2銆佷护u=tanx锛岃繘琛屼笁瑙掍唬鎹紝灏嗗叾绠鍖栵紝鍐嶆寜鍩烘湰绉垎鍏紡杩涜璁＄畻銆3銆佸皢鍙橀噺鍥炰唬锛屾渶鍚庡緱鍒伴棶棰樼殑缁撴灉 銆愭眰瑙ｈ繃绋嬨戙愭湰棰樼煡璇嗙偣銆1銆佷笉瀹氱Н鍒嗐傝f(x)鍦ㄦ煇鍖洪棿I涓婃湁瀹氫箟锛屽鏋滃瓨鍦ㄥ嚱鏁癋(x)锛屼娇寰楀浜庝换...
绛旓細3. 浠ｆ崲鍨3锛氬綋鍑虹幇褰㈠ x^2 - a^2 鐨勫钩鏂规牴鏃讹紝鍙互浣跨敤浠ｆ崲 x = a * sec(胃)銆4. 浠ｆ崲鍨4锛氬綋鍑虹幇褰㈠ x^2 + a^2 鐨勫钩鏂规牴鏃讹紝鍙互浣跨敤浠ｆ崲 x = a * cot(胃)銆傞氳繃杩欎簺浠ｆ崲鍏紡锛屾垜浠彲浠ュ皢鍘熷鐨勪笁瑙掑嚱鏁扮Н鍒杞寲涓烘洿绠鍗曠殑涓夎鍑芥暟绉垎鎴栧熀鏈殑甯告暟绉垎銆傜劧鍚庯紝閫氳繃璁＄畻寰楀埌绉...
绛旓細4.鍒╃敤瀵圭О鎬э細瀵逛簬鏌愪簺涓夎鍑芥暟锛屾垜浠彲浠ュ埄鐢ㄥ畠浠殑瀵圭О鎬ф潵绠鍖栫Н鍒嗙殑璁＄畻銆備緥濡傦紝姝ｅ鸡鍑芥暟鍜屼綑寮﹀嚱鏁伴兘鏄叧浜巠杞村绉扮殑锛屾墍浠ユ垜浠彲浠ュ皢绉垎鍖洪棿鍒嗕负涓や釜閮ㄥ垎锛岀劧鍚庡垎鍒绠楁瘡涓儴鍒嗙殑绉垎锛屾渶鍚庡皢杩欎袱涓Н鍒嗙浉鍔犲緱鍒版荤殑绉垎鍊笺5.鍒╃敤涓夎鎭掔瓑寮忥細鍦ㄨ绠椾笁瑙掑嚱鏁扮殑绉垎鏃讹紝鎴戜滑杩樺彲浠ュ埄鐢ㄤ竴浜涗笁瑙掓亽绛...
绛旓細涓夎鍑芥暟姹傜Н鍒涓囪兘鍏紡锛氾紙sin伪锛塣2+锛坈os伪锛塣2=1锛1+锛坱an伪锛塣2=锛坰ec伪锛塣2锛1+锛坈ot伪锛塣2=锛坈sc伪锛塣2銆備笁瑙掑嚱鏁版槸鍩烘湰鍒濈瓑鍑芥暟涔嬩竴锛屾槸浠ヨ搴︿负鑷彉閲忥紝瑙掑害瀵瑰簲浠绘剰瑙掔粓杈逛笌鍗曚綅鍦嗕氦鐐瑰潗鏍囨垨鍏舵瘮鍊间负鍥犲彉閲忕殑鍑芥暟銆備篃鍙互绛変环鍦扮敤涓庡崟浣嶅渾鏈夊叧鐨勫悇绉嶇嚎娈电殑闀垮害鏉ュ畾涔夈備笁瑙掑嚱鏁板湪...
绛旓細姝ら鍏抽敭鏄垎姝绉垎娉曞拰涓夎鍑芥暟鐨闄嶉樁绛夈傚垎閮ㄧН鍒嗘硶 璁惧嚱鏁板拰u锛寁鍏锋湁杩炵画瀵兼暟锛屽垯d(uv)=udv+vdu銆傜Щ椤瑰緱鍒皍dv=d(uv)-vdu 涓よ竟绉垎锛屽緱鍒嗛儴绉垎鍏紡 鈭玼dv=uv-鈭玽du銆 鈶 绉板叕寮忊懘涓哄垎閮ㄧН鍒嗗叕寮忥紟濡傛灉绉垎鈭玽du鏄撲簬姹傚嚭锛屽垯宸︾绉垎寮忛殢涔嬪緱鍒帮紟鍒嗛儴绉垎鍏紡杩愮敤鎴愯触鐨勫叧閿槸鎭板綋鍦伴夋嫨u,v ...
绛旓細3. 鍙嶆鍒囧嚱鏁帮細$\int \arctan(x) \, dx = x \arctan(x) - \frac{1}{2} \ln(1 + x^2) + C 杩欓噷锛$C$ 鏄Н鍒嗗父鏁帮紝琛ㄧず涓嶅畾绉垎鐨勫父鏁伴儴鍒嗐傚鏋滀綘閬囧埌鏇村鏉傜殑鍙涓夎鍑芥暟鐨勭Н鍒锛屽彲鑳介渶瑕佷娇鐢ㄤ竴浜涙洿楂樼骇鐨勭Н鍒嗘妧宸锛屽閮ㄥ垎绉垎銆佹崲鍏冩硶绛夈傚湪浣跨敤杩欎簺鍏紡鏃讹紝鏈閲嶈鐨勬槸纭繚浣...
绛旓細\int \csc^2(x) , dx = -\cot(x) + C \int \sec(x) \tan(x) , dx = \sec(x) + C \int \csc(x) \cot(x) , dx = -\csc(x) + C 杩欎簺鏄竴浜涘熀鏈殑涓夎鍑芥暟绉垎鍏紡銆傚鏋滈渶瑕佸鏇村鏉傜殑涓夎鍑芥暟杩涜绉垎锛屽彲浠ヤ娇鐢绉垎鎶宸锛屽閮ㄥ垎绉垎銆佷笁瑙掓亽绛夊紡銆佷笁瑙掓浛鎹㈢瓑銆傚悓鏃讹紝...
绛旓細瀵逛簬鍑芥暟 f(x) = -1/2xcos2x - 1/4sin2x + xcosx + sinx锛屽湪鍖洪棿 (0, 蟺) 涓姹傜Н鍒锛屽叾琚Н鍑芥暟鍖呭惈浜嗕笁瑙掑嚱鏁帮紝鍥犳鎴戜滑鍙互鑰冭檻浣跨敤涓夎鍑芥暟鐨鎬ц川鏉ョ畝鍖栫Н鍒嗐傞鍏堬紝鎴戜滑鍙互灏嗗嚱鏁 f(x) 鍐欐垚 f(x) = -1/2x(2cos^2x - 1) - 1/4sin^2x + xcosx + sinx = -1/2xcos^2x...
