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Integral with Trig Functions

83​

∫sin⁡x dx=−cos⁡x+C\int{\sin{x}}\ \mathrm{d}x=-\cos{x}+C
Derivation

84​

∫cos⁡x dx=sin⁡x+C\int{\cos{x}}\ \mathrm{d}x=\sin{x}+C
Derivation

85​

∫tan⁡x dx=−ln⁡∣cos⁡x∣+C\int{\tan{x}}\ \mathrm{d}x=-\ln{\left|\cos{x} \right|}+C
Derivation

86​

∫cot⁡x dx=ln⁡∣sin⁡x∣+C\int{\cot{x}}\ \mathrm{d}x=\ln{\left|\sin{x} \right|}+C
Derivation

87​

∫sec⁡x dx=ln⁡∣tan⁡(π4+x2)∣+C=ln⁡∣sec⁡x+tan⁡x∣+C\int{\sec{x}}\ \mathrm{d}x=\ln{\left|\tan{\left(\frac{\pi}{4}+\frac{x}{2} \right)} \right|}+C=\ln{\left|\sec{x}+\tan{x} \right|}+C
Derivation

88​

∫csc⁡x dx=ln⁡∣tan⁡x2∣+C=ln⁡∣csc⁡x−cot⁡x∣+C\int{\csc{x}}\ \mathrm{d}x=\ln{\left|\tan{\frac{x}{2}} \right|}+C=\ln{\left|\csc{x}-\cot{x} \right|}+C
Derivation

89​

∫sec⁡2x dx=tan⁡x+C\int{\sec^2{x}}\ \mathrm{d}x=\tan{x}+C
Derivation

90​

∫csc⁡2x dx=−cot⁡x+C\int{\csc^2{x}}\ \mathrm{d}x=-\cot{x}+C
Derivation

91​

∫sec⁡xtan⁡x dx=sec⁡x+C\int{\sec{x}\tan{x}}\ \mathrm{d}x=\sec{x}+C
Derivation

92​

∫csc⁡xcot⁡x dx=−csc⁡x+C\int{\csc{x}\cot{x}}\ \mathrm{d}x=-\csc{x}+C
Derivation

93​

∫sin⁡2x dx=x2−14sin⁡2x+C\int{\sin^2{x}}\ \mathrm{d}x=\frac{x}{2}-\frac{1}{4}\sin{2x}+C
Derivation

94​

∫cos⁡2x dx=x2+14sin⁡2x+C\int{\cos^2{x}}\ \mathrm{d}x=\frac{x}{2}+\frac{1}{4}\sin{2x}+C
Derivation

95​

∫sin⁡nx dx=−1nsin⁡n−1xcos⁡x+n−1n∫sin⁡n−2x dx\int{\sin^n{x}}\ \mathrm{d}x=-\frac{1}{n}\sin^{n-1}{x}\cos{x}+\frac{n-1}{n}\int{\sin^{n-2}{x}}\ \mathrm{d}x
Derivation

96​

∫cos⁡nx dx=1ncos⁡n−1xsin⁡x+n−1n∫cos⁡n−2x dx\int{\cos^n{x}}\ \mathrm{d}x=\frac{1}{n}\cos^{n-1}{x}\sin{x}+\frac{n-1}{n}\int{\cos^{n-2}{x}}\ \mathrm{d}x
Derivation

97​

∫1sin⁡nx dx=−1n−1⋅cos⁡xsin⁡n−1x+n−2n−1∫1sin⁡n−2x dx\int{\frac{1}{\sin^n{x}}}\ \mathrm{d}x=-\frac{1}{n-1}\cdot \frac{\cos{x}}{\sin^{n-1}{x}}+\frac{n-2}{n-1}\int{\frac{1}{\sin^{n-2}{x}}}\ \mathrm{d}x
Derivation

98​

∫1cos⁡nx dx=1n−1⋅sin⁡xcos⁡n−1x+n−2n−1∫1cos⁡n−2x dx\int{\frac{1}{\cos^n{x}}}\ \mathrm{d}x=\frac{1}{n-1}\cdot \frac{\sin{x}}{\cos^{n-1}{x}}+\frac{n-2}{n-1}\int{\frac{1}{\cos^{n-2}{x}}}\ \mathrm{d}x
Derivation

99​

∫cos⁡mxsin⁡nx dx=1m+ncos⁡m−1xsin⁡n+1x+m−1m+n∫cos⁡m−2xsin⁡nx dx=−1m+ncos⁡m+1xsin⁡n−1x+n−1m+n∫cos⁡mxsin⁡n−2x dx\begin{align*} \int{\cos^{m}{x}\sin^{n}{x}}\ \mathrm{d}x&=\frac{1}{m+n}\cos^{m-1}{x}\sin^{n+1}{x}+\frac{m-1}{m+n}\int{\cos^{m-2}{x}\sin^{n}{x}}\ \mathrm{d}x\\ &=-\frac{1}{m+n}\cos^{m+1}{x}\sin^{n-1}{x}+\frac{n-1}{m+n}\int{\cos^{m}{x}\sin^{n-2}{x}}\ \mathrm{d}x \end{align*}
Derivation

100​

∫sin⁡axcos⁡bx dx=−12(a+b)cos⁡(a+b)x−12(a−b)cos⁡(a−b)x+C\int{\sin{ax}\cos{bx}}\ \mathrm{d}x=-\frac{1}{2(a+b)}\cos{(a+b)x}-\frac{1}{2(a-b)}\cos{(a-b)x}+C
Derivation

101​

∫sin⁡axsin⁡bx dx=−12(a+b)sin⁡(a+b)x+12(a−b)sin⁡(a−b)x+C\int{\sin{ax}\sin{bx}}\ \mathrm{d}x=-\frac{1}{2(a+b)}\sin{(a+b)x}+\frac{1}{2(a-b)}\sin{(a-b)x}+C
Derivation

102​

∫cos⁡axcos⁡bx dx=12(a+b)sin⁡(a+b)x+12(a−b)sin⁡(a−b)x+C\int{\cos{ax}\cos{bx}}\ \mathrm{d}x=\frac{1}{2(a+b)}\sin{(a+b)x}+\frac{1}{2(a-b)}\sin{(a-b)x}+C
Derivation

103​

∫1a+bsin⁡x dx=2a2−b2arctan⁡aarctan⁡x2+ba2−b2+C(a2>b2)\int{\frac{1}{a+b\sin{x}}}\ \mathrm{d}x=\frac{2}{\sqrt{a^2-b^2}}\arctan{\frac{a\arctan{\frac{x}{2}}+b}{\sqrt{a^2-b^2}}}+C \quad (a^2>b^2)
Derivation

104​

∫1a+bsin⁡x dx=1a2−b2ln⁡∣atan⁡x2+b−b2−a2atan⁡x2+b+b2−a2∣+C(a2<b2)\int{\frac{1}{a+b\sin{x}}}\ \mathrm{d}x=\frac{1}{\sqrt{a^2-b^2}}\ln{\left|\frac{a\tan{\frac{x}{2}}+b-\sqrt{b^2-a^2}}{a\tan{\frac{x}{2}+b+\sqrt{b^2-a^2}}}\right|} +C \quad(a^2<b^2)
Derivation

105​

∫1a+bcos⁡x dx=2a+ba+ba−barctan⁡(a−ba+btan⁡x2)+C(a2>b2)\int{\frac{1}{a+b\cos{x}}}\ \mathrm{d}x=\frac{2}{a+b}\sqrt{\frac{a+b}{a-b}}\arctan{\left(\sqrt{\frac{a-b}{a+b}}\tan{\frac{x}{2}}\right)}+C \quad (a^2>b^2)
Derivation

106​

∫1a+bcos⁡x dx=1a+ba+ba−bln⁡∣tan⁡x2+a+bb−atan⁡x2−a+bb−a∣+C(a2<b2)\int{\frac{1}{a+b\cos{x}}}\ \mathrm{d}x=\frac{1}{a+b}\sqrt{\frac{a+b}{a-b}}\ln{\left|\frac{\tan{\frac{x}{2}}+\sqrt{\frac{a+b}{b-a}}}{\tan{\frac{x}{2}-\sqrt{\frac{a+b}{b-a}}}}\right|} +C \quad(a^2<b^2)
Derivation

107​

∫1a2cos⁡2x+b2sin⁡2x dx=1abarctan⁡(batan⁡x)+C\int{\frac{1}{a^2\cos^2{x}+b^2\sin^2{x}}}\ \mathrm{d}x=\frac{1}{ab}\arctan{\left(\frac{b}{a}\tan{x}\right)}+C
Derivation

108​

∫1a2cos⁡2x−b2sin⁡2x dx=12abln⁡∣btan⁡x+abtan⁡x−a∣+C\int{\frac{1}{a^2\cos^2{x}-b^2\sin^2{x}}}\ \mathrm{d}x=\frac{1}{2ab}\ln{\left|\frac{b\tan{x}+a}{b\tan{x}-a} \right|}+C
Derivation

109​

∫xsin⁡ax dx=1a2sin⁡ax−1axcos⁡ax+C\int{x\sin{ax}}\ \mathrm{d}x=\frac{1}{a^2}\sin{ax}-\frac{1}{a}x\cos{ax}+C
Derivation

110​

∫x2sin⁡ax dx=−1a2x2cos⁡ax−2a2xsin⁡ax+2a3cos⁡ax+C\int{x^2\sin{ax}}\ \mathrm{d}x=-\frac{1}{a^2}x^2\cos{ax}-\frac{2}{a^2}x\sin{ax}+\frac{2}{a^3}\cos{ax}+C
Derivation

111​

∫xcos⁡ax dx=1a2sin⁡ax+1axsin⁡ax+C\int{x\cos{ax}}\ \mathrm{d}x=\frac{1}{a^2}\sin{ax}+\frac{1}{a}x\sin{ax}+C
Derivation

112​

∫x2cos⁡ax dx=1a2x2sin⁡ax+2a2xcos⁡ax−2a3sin⁡ax+C\int{x^2\cos{ax}}\ \mathrm{d}x=\frac{1}{a^2}x^2\sin{ax}+\frac{2}{a^2}x\cos{ax}-\frac{2}{a^3}\sin{ax}+C
Derivation