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Integration Formulas:

Integration of some standard functions
Integration of \(x^n\) with respect to \(x\) (dx) ∫ xⁿdx = ——xⁿ⁺¹𝑛+1 + c   , n≠ -1 www.faastop.comاللهπ∫ xⁿdx = ——xⁿ⁺¹𝑛+1 + c   , n≠ -1 ∫ xⁿdx = ——xⁿ⁺¹𝑛+1 + c   , n≠ -1
Integration of \( \frac{1}{x} \) with respect to \(x\) (dx) ∫——1x dx = 𝑙𝑜𝑔ₑ|x| + cwww.faastop.comاللهπ∫——1x dx = 𝑙𝑜𝑔ₑ|x| + c∫——1x dx = 𝑙𝑜𝑔ₑ|x| + c
Integration of \(e^x\) with respect to \(x\) (dx) ∫𝑒x dx = 𝑒x + cwww.faastop.comاللهπ∫𝑒x dx = 𝑒x + c∫𝑒x dx = 𝑒x + c
Integration_of_a_power_x
Integration of Trigonometric Functions
Integration of \( \sin(x) \) with respect to \( x \) (dx) ∫𝑠𝑖𝑛x dx = - 𝑐𝑜𝑠x + cwww.faastop.comاللهπ∫𝑠𝑖𝑛x dx = - 𝑐𝑜𝑠x + c∫𝑠𝑖𝑛x dx = - 𝑐𝑜𝑠x + c
Integration of \( \cos(x) \) with respect to \( x \) (dx) ∫𝑐𝑜𝑠x dx = 𝑠𝑖𝑛x + cwww.faastop.comاللهπ∫𝑐𝑜𝑠x dx = 𝑠𝑖𝑛x + c∫𝑐𝑜𝑠x dx = 𝑠𝑖𝑛x + c
Integration of \( \tan(x) \) with respect to \( x \) (dx) ∫𝑡𝑎𝑛x dx = - 𝑙𝑜𝑔|𝑐𝑜𝑠x| + cwww.faastop.comاللهπ∫𝑡𝑎𝑛x dx = - 𝑙𝑜𝑔|𝑐𝑜𝑠x| + c∫𝑡𝑎𝑛x dx = - 𝑙𝑜𝑔|𝑐𝑜𝑠x| + c
Integration of \( \cot(x) \) with respect to \( x \) (dx) ∫𝑐𝑜𝑡x dx = 𝑙𝑜𝑔|𝑠𝑖𝑛x| + cwww.faastop.comاللهπ∫𝑐𝑜𝑡x dx = 𝑙𝑜𝑔|𝑠𝑖𝑛x| + c∫𝑐𝑜𝑡x dx = 𝑙𝑜𝑔|𝑠𝑖𝑛x| + c
Integration of \( \sec(x) \) with respect to \( x \) (dx) ∫𝑠𝑒𝑐x dx = 𝑙𝑜𝑔|𝑠𝑒𝑐x+𝑡𝑎𝑛x| + cwww.faastop.comاللهπ∫𝑠𝑒𝑐x dx = 𝑙𝑜𝑔|𝑠𝑒𝑐x+𝑡𝑎𝑛x| + c∫𝑠𝑒𝑐x dx = 𝑙𝑜𝑔|𝑠𝑒𝑐x+𝑡𝑎𝑛x| + c
Integration of \( \csc(x) \) with respect to \( x \) (dx) ∫𝑐𝑜𝑠𝑒𝑐x dx = 𝑙𝑜𝑔|𝑐𝑜𝑠𝑒𝑐x-𝑐𝑜𝑡x| + cwww.faastop.comاللهπ∫𝑐𝑜𝑠𝑒𝑐x dx = 𝑙𝑜𝑔|𝑐𝑜𝑠𝑒𝑐x-𝑐𝑜𝑡x| + c∫𝑐𝑜𝑠𝑒𝑐x dx = 𝑙𝑜𝑔|𝑐𝑜𝑠𝑒𝑐x-𝑐𝑜𝑡x| + c
Integration of \( \sec^2(x) \) with respect to \( x \) (dx) ∫𝑠𝑒𝑐²x dx = 𝑡𝑎𝑛x + 𝑐 www.faastop.comاللهπ∫𝑠𝑒𝑐²x dx = 𝑡𝑎𝑛x + 𝑐 ∫𝑠𝑒𝑐²x dx = 𝑡𝑎𝑛x + 𝑐
Integration of \( \csc^2(x) \) with respect to \( x \) (dx) ∫𝑐𝑜𝑠𝑒𝑐²x dx = - 𝑐𝑜𝑡x + 𝑐www.faastop.comاللهπ∫𝑐𝑜𝑠𝑒𝑐²x dx = - 𝑐𝑜𝑡x + 𝑐∫𝑐𝑜𝑠𝑒𝑐²x dx = - 𝑐𝑜𝑡x + 𝑐
Integration of \( \sec(x) \tan(x) \) with respect to \( x \) (dx) ∫𝑠𝑒𝑐x 𝑡𝑎𝑛x dx = 𝑠𝑒𝑐x + cwww.faastop.comاللهπ∫𝑠𝑒𝑐x 𝑡𝑎𝑛x dx = 𝑠𝑒𝑐x + c∫𝑠𝑒𝑐x 𝑡𝑎𝑛x dx = 𝑠𝑒𝑐x + c
Integration of \( \csc(x) \cot(x) \) with respect to \( x \) (dx) ∫𝑐𝑜𝑠𝑒𝑐x 𝑐𝑜𝑡x dx = - 𝑐𝑜𝑠𝑒𝑐x + cwww.faastop.comاللهπ∫𝑐𝑜𝑠𝑒𝑐x 𝑐𝑜𝑡x dx = - 𝑐𝑜𝑠𝑒𝑐x + c∫𝑐𝑜𝑠𝑒𝑐x 𝑐𝑜𝑡x dx = - 𝑐𝑜𝑠𝑒𝑐x + c
Integration of Hyperbolic Functions
Integration of \( \sinh(x) \) with respect to \( x \) (dx) ∫𝑠𝑖𝑛𝒉x dx = 𝑐𝑜𝑠𝒉x + cwww.faastop.comاللهπ∫𝑠𝑖𝑛𝒉x dx = 𝑐𝑜𝑠𝒉x + c∫𝑠𝑖𝑛𝒉x dx = 𝑐𝑜𝑠𝒉x + c
Integration of \( \cosh(x) \) with respect to \( x \) (dx) ∫𝑐𝑜𝑠𝒉x dx = 𝑠𝑖𝑛𝒉x + cwww.faastop.comاللهπ∫𝑐𝑜𝑠𝒉x dx = 𝑠𝑖𝑛𝒉x + c∫𝑐𝑜𝑠𝒉x dx = 𝑠𝑖𝑛𝒉x + c
Integration of \( \tanh(x) \) with respect to \( x \) (dx) ∫𝑡𝑎𝑛𝒉x dx = 𝑙𝑜𝑔|𝑐𝑜𝑠𝒉x| + cwww.faastop.comاللهπ∫𝑡𝑎𝑛𝒉x dx = 𝑙𝑜𝑔|𝑐𝑜𝑠𝒉x| + c∫𝑡𝑎𝑛𝒉x dx = 𝑙𝑜𝑔|𝑐𝑜𝑠𝒉x| + c
Integration of \( \coth(x) \) with respect to \( x \) (dx) ∫𝑐𝑜𝑡𝒉x dx = 𝑙𝑜𝑔|𝑠𝑖𝑛𝒉x| + cwww.faastop.comاللهπ∫𝑐𝑜𝑡𝒉x dx = 𝑙𝑜𝑔|𝑠𝑖𝑛𝒉x| + c∫𝑐𝑜𝑡𝒉x dx = 𝑙𝑜𝑔|𝑠𝑖𝑛𝒉x| + c
Integration of \( \text{sech}(x) \) with respect to \( x \) (dx) ∫𝑠𝑒𝑐𝒉x dx = 𝑡𝑎𝑛⁻¹(𝑠𝑖𝑛𝒉x) + cwww.faastop.comاللهπ∫𝑠𝑒𝑐𝒉x dx = 𝑡𝑎𝑛⁻¹(𝑠𝑖𝑛𝒉x) + c∫𝑠𝑒𝑐𝒉x dx = 𝑡𝑎𝑛⁻¹(𝑠𝑖𝑛𝒉x) + c
Integration of \( \text{csch}(x) \) with respect to \( x \) (dx) ∫𝑐𝑜𝑠𝑒𝑐𝒉x dx = 𝑙𝑜𝑔|𝑡𝑎𝑛𝒉 ——x2 | + cwww.faastop.comاللهπ∫𝑐𝑜𝑠𝑒𝑐𝒉x dx = 𝑙𝑜𝑔|𝑡𝑎𝑛𝒉 ——x2 | + c∫𝑐𝑜𝑠𝑒𝑐𝒉x dx = 𝑙𝑜𝑔|𝑡𝑎𝑛𝒉 ——x2 | + c
Integration of \( \text{sech}^2(x) \) with respect to \( x \) (dx) ∫𝑠𝑒𝑐𝒉²x dx = 𝑡𝑎𝑛𝒉x + cwww.faastop.comاللهπ∫𝑠𝑒𝑐𝒉²x dx = 𝑡𝑎𝑛𝒉x + c∫𝑠𝑒𝑐𝒉²x dx = 𝑡𝑎𝑛𝒉x + c
Integration of \( \text{csch}^2(x) \) with respect to \( x \) (dx) ∫𝑐𝑜𝑠𝑒𝑐𝒉²x dx = -𝑐𝑜𝑡𝒉x + cwww.faastop.comاللهπ∫𝑐𝑜𝑠𝑒𝑐𝒉²x dx = -𝑐𝑜𝑡𝒉x + c∫𝑐𝑜𝑠𝑒𝑐𝒉²x dx = -𝑐𝑜𝑡𝒉x + c
Integration of \( \text{sech}(x) \tanh(x) \) with respect to \( x \) (dx) ∫𝑠𝑒𝑐𝒉x 𝑡𝑎𝑛𝒉x dx = -𝑠𝑒𝑐𝒉x + cwww.faastop.comاللهπ∫𝑠𝑒𝑐𝒉x 𝑡𝑎𝑛𝒉x dx = -𝑠𝑒𝑐𝒉x + c∫𝑠𝑒𝑐𝒉x 𝑡𝑎𝑛𝒉x dx = -𝑠𝑒𝑐𝒉x + c
Integration of \( \text{csch}(x) \coth(x) \) with respect to \( x \) (dx)∫𝑐𝑜𝑠𝑒𝑐𝒉x 𝑐𝑜𝑡𝒉x dx = -𝑐𝑜𝑠𝑒𝑐𝒉x + cwww.faastop.comاللهπ∫𝑐𝑜𝑠𝑒𝑐𝒉x 𝑐𝑜𝑡𝒉x dx = -𝑐𝑜𝑠𝑒𝑐𝒉x + c∫𝑐𝑜𝑠𝑒𝑐𝒉x 𝑐𝑜𝑡𝒉x dx = -𝑐𝑜𝑠𝑒𝑐𝒉x + c
Integration of Some other Functions
Integration of \( \frac{1}{\sqrt{a^2 - x^2}} \) with respect to \( x \) (dx) ∫1𝑎²-x² 𝑑x = 𝑠𝑖𝑛 -1[——x𝑎] + cwww.faastop.comاللهπ∫1𝑎²-x² 𝑑x = 𝑠𝑖𝑛 -1[——x𝑎] + c∫1𝑎²-x² 𝑑x = 𝑠𝑖𝑛 -1[——x𝑎] + c
Integration of \( -\frac{1}{\sqrt{a^2 - x^2}} \) with respect to \( x \) (dx) ∫-1𝑎²-x²  𝑑x = 𝑐𝑜𝑠 -1[——x𝑎] + cwww.faastop.comاللهπ∫-1𝑎²-x²  𝑑x = 𝑐𝑜𝑠 -1[——x𝑎] + c∫-1𝑎²-x²  𝑑x = 𝑐𝑜𝑠 -1[——x𝑎] + c
Integration of \( \frac{1}{a^2 + x^2} \) with respect to \( x \) (dx) ∫ 1𝑎²+x² 𝑑x = ——1𝑎 𝑡𝑎𝑛 -1[——x𝑎] + c www.faastop.comاللهπ∫ 1𝑎²+x² 𝑑x = ——1𝑎 𝑡𝑎𝑛 -1[——x𝑎] + c ∫ 1𝑎²+x² 𝑑x = ——1𝑎 𝑡𝑎𝑛 -1[——x𝑎] + c
Integration of \( -\frac{1}{a^2 + x^2} \) with respect to \( x \) (dx) ∫-1𝑎²+x² 𝑑x = ——1𝑎 𝑐𝑜𝑡 -1[——x𝑎] + cwww.faastop.comاللهπ∫-1𝑎²+x² 𝑑x = ——1𝑎 𝑐𝑜𝑡 -1[——x𝑎] + c∫-1𝑎²+x² 𝑑x = ——1𝑎 𝑐𝑜𝑡 -1[——x𝑎] + c
Integration of \( -\frac{1}{a^2 - x^2} \) with respect to \( x \) (dx) ∫1x  x²-𝑎² 𝑑x = ——1𝑎 𝑠𝑒𝑐 -1[——x𝑎] + cwww.faastop.comاللهπ∫1x  x²-𝑎² 𝑑x = ——1𝑎 𝑠𝑒𝑐 -1[——x𝑎] + c∫1x  x²-𝑎² 𝑑x = ——1𝑎 𝑠𝑒𝑐 -1[——x𝑎] + c
Integration of \( -\frac{1}{x \sqrt{x^2 - a^2}} \) with respect to \( x \) (dx) ∫ -1x  x²-𝑎² 𝑑x = ——1𝑎 𝑐𝑜𝑠𝑒𝑐 -1[——x𝑎] + cwww.faastop.comاللهπ∫ -1x  x²-𝑎² 𝑑x = ——1𝑎 𝑐𝑜𝑠𝑒𝑐 -1[——x𝑎] + c∫ -1x  x²-𝑎² 𝑑x = ——1𝑎 𝑐𝑜𝑠𝑒𝑐 -1[——x𝑎] + c

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