∫ dx
α^2 −β^2 sin^2 ax=
1
aα√
α^2 −β^2tan−^1(√
α^2 −β^2
α tanax)
+C, α^2 > β^21
2 aα√
β^2 −α^2ln∣∣
∣∣
∣√
√β^2 −α^2 tanax+α
β^2 −α^2 tanax−α∣∣
∣∣
∣+C, α(^2) < β 2
417.
∫
sinnax dx=−an^1 sinn−^1 axcosax+n−n^1
∫
sinn−^2 ax dx
418.
∫ dx
sinnax=
−cosax
(n−1)asinn−^1 ax+
n− 2
n− 1
∫ dx
sinn−^2 ax
419.
∫
xnsinax dx=−a^1 xncosax+na
∫
xn−^1 cosax dx
420.
∫ α+βsinax
1 ±sinax dx=βx+
α∓β
a tan
(π
4 ∓
ax
2
)
+C
421.
∫ α+βsinax
a+bsinaxdx=
β
bx+
αb−aβ
b
∫ dx
a+bsinax
422.
∫ dx
α+sinβax
=xα−βα
∫ dx
β+αsinax
Integrals containing cosax
423.
∫
cosax dx=^1 asinax+C
424.
∫
xcosax dx=a^12 cosax+xasinax+C
425.
∫
x^2 cosax dx=^2 ax 2 cosax+
(
x^2
a −
2
a^3
)
sinax+C
426.
∫
xncosax dx=^1 axnsinax+an 2 xn−^1 cosax−n(na− 2 1)
∫
xn−^2 cosax dx
427.
∫ cosax
x dx= ln|x|−
a^2 x^2
2 ·2!+
a^4 x^4
4 ·4!−
a^6 x^6
6 ·6!+···+
(−1)na^2 nx^2 n
(2n)·(2n)! +···+C
428.
∫ cosax dx
xn =−
cosax
(n−1)xn−^1 −
a
n− 1
∫ sinax
xn−^1 dx
429.
∫ dx
cosax=
1
aln|secax+ tanax|+C
430.
∫ x dx
cosax=
1
a^2
[
a^2 x^2
2 +
a^4 x^4
4 ·2!+
5 a^6 x^6
6 ·4! +···+
Ena^2 n+2x^2 n+2
(2n+ 2)·(2n)!+···
]
+C
431.
∫ dx
xcosax= ln|x|+
a^2 x^2
4 +
5 a^4 x^4
96 +···+
Ena^2 nx^2 n
2 n(2n)! +···+C
whereEnis the nthEuler numberE 1 = 1,E 2 = 5,E 3 = 61,.. .Note scaling and shifting
Appendix C